Number Name How many
0 zero 1 one
2 two 3 three
4 four 5 five
6 six 7 seven
8 eight 9 nine
10 ten 20 twenty two tens
30 thirty three tens 40 forty four tens
50 fifty five tens 60 sixty six tens
70 seventy seven tens 80 eighty eight tens 90 ninety nine tens
Number Name How Many
100 one hundred ten tens 1,000 one thousand ten hundreds
10,000 ten thousand ten thousands 100,000 one hundred thousand one hundred thousands
1,000,000 one million one thousand thousands
Fractions
Digits to the right of the decimal point represent the fractional part of the decimal number. Each place value has a value that is one tenth the value to the immediate left of it.
Number Name Fraction
.1 tenth 1/10
.01 hundredth 1/100
.001 thousandth 1/1000
.0001 ten thousandth 1/10000
.00001 hundred thousandth 1/100000
Examples:
0.234 = 234/1000 (said - point 2 3 4, or 234 thousandths, or two hundred thirty four thousandths)
4.83 = 4 83/100 (said - 4 point 8 3, or 4 and 83 hundredths)
SI Prefixes
Number Prefix Symbol
10 1 deka- da 10 2 hecto- h
10 3 kilo- k 10 6 mega- M
10 9 giga- G 10 12 tera- T
10 15 peta- P 10 18 exa- E
10 21 zeta- Z 10 24 yotta- Y
10 -1 deci- d 10 -2 centi- c
10 -3 milli- m 10 -6 micro- u (greek mu)
10 -9 nano- n 10 -12 pico- p
10 -15 femto- f 10 -18 atto- a
10 -21 zepto- z 10 -24 yocto- y
Roman Numerals
I=1 (I with a bar is not used)
V=5 _ V=5,000
X=10 _ X=10,000
L=50 _ L=50,000
C=100 _ C = 100 000
D=500 _ D=500,000
M=1,000 _
M=1,000,000
Roman Numeral Calculator
Examples:
1 = I 2 = II 3 = III 4 = IV
5 = V 6 = VI 7 = VII 8 = VIII
9 = IX 10 = X 11 = XI 12 = XII
13 = XIII 14 = XIV 15 = XV 16 = XVI
17 = XVII 18 = XVIII 19 = XIX 20 = XX
21 = XXI 25 = XXV 30 = XXX 40 = XL
49 = XLIX 50 = L 51 = LI 60 = LX
70 = LXX 80 = LXXX 90 = XC 99 = XCIX
There is no zero in the roman numeral system.
The numbers are built starting from the largest number on the left, and adding smaller numbers to the right. All the numerals are then added together.
The exception is the subtracted numerals, if a numeral is before a larger numeral, you subtract the first numeral from the second. That is, IX is 10 - 1= 9.
This only works for one small numeral before one larger numeral - for example, IIX is not 8, it is not a recognized roman numeral.
There is no place value in this system - the number III is 3, not 111.
Number Base Systems Decimal(10) Binary(2) Ternary(3) Octal(8) Hexadecimal(16)
0
0 0 0 0
1 1 1 1 1
2 10 2 2 2
3 11 10 3 3
4 100 11 4 4
5 101 12 5 5
6 110 20 6 6
7 111 21 7 7
8 1000 22 10 8
9 1001 100 11 9
10 1010 101 12 A
11 1011 102 13 B
12 1100 110 14 C
13 1101 111 15 D
14 1110 112 16 E
15 1111 120 17 F
16 10000 121 20 10
17 10001 122 21 11
18 10010 200 22 12
19 10011 201 23 13
20 10100 202 24 14
Each digit can only count up to the value of one less than the base. In hexadecimal, the letters A - F are used to represent the digits 10 - 15, so they would only use one character.
Addition Table
+ 0 1 2 3 4 5 6 7 8 9 10
0 0 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10 11
2 2 3 4 5 6 7 8 9 10 11 12
3 3 4 5 6 7 8 9 10 11 12 13
4 4 5 6 7 8 9 10 11 12 13 14
5 5 6 7 8 9 10 11 12 13 14 15
6 6 7 8 9 10 11 12 13 14 15 16
7 7 8 9 10 11 12 13 14 15 16 17
8 8 9 10 11 12 13 14 15 16 17 18
9 9 10 11 12 13 14 15 16 17 18 19
10 10 11 12 13 14 15 16 17 18 19 20
Alternative Format
x 0 1 2 3 4 5 6 7 8 9 10 11 12
0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11 12
2 0 2 4 6 8 10 12 14 16 18 20 22 24
3 0 3 6 9 12 15 18 21 24 27 30 33 36
4 0 4 8 12 16 20 24 28 32 36 40 44 48
5 0 5 10 15 20 25 30 35 40 45 50 55 60
6 0 6 12 18 24 30 36 42 48 54 60 66 72
7 0 7 14 21 28 35 42 49 56 63 70 77 84
8 0 8 16 24 32 40 48 56 64 72 80 88 96
9 0 9 18 27 36 45 54 63 72 81 90 99 108
10 0 10 20 30 40 50 60 70 80 90 100 110 120
11 0 11 22 33 44 55 66 77 88 99 110 121 132
12 0 12 24 36 48 60 72 84 96 108 120 132 144
Fraction to Decimal Conversion Tables
Important Note: any span of numbers that is underlined signifies that those numbers are repeated. For example, 0.09 signifies 0.090909....
Only fractions in lowest terms are listed. For instance, to find 2/8, first simplify it to 1/4 then search for it in the table below. fraction = decimal
1/1 = 1
1/2 = 0.5
1/3 = 0.3 2/3 = 0.6
1/4 = 0.25 3/4 = 0.75
1/5 = 0.2 2/5 = 0.4 3/5 = 0.6 4/5 = 0.8
1/6 = 0.16 5/6 = 0.83
1/7 = 0.142857 2/7 = 0.285714 3/7 = 0.428571 4/7 = 0.571428
5/7 = 0.714285 6/7 = 0.857142
1/8 = 0.125 3/8 = 0.375 5/8 = 0.625 7/8 = 0.875
1/9 = 0.1 2/9 = 0.2 4/9 = 0.4 5/9 = 0.5
7/9 = 0.7 8/9 = 0.8
1/10 = 0.1 3/10 = 0.3 7/10 = 0.7 9/10 = 0.9
1/11 = 0.09 2/11 = 0.18 3/11 = 0.27 4/11 = 0.36
5/11 = 0.45 6/11 = 0.54 7/11 = 0.63
8/11 = 0.72 9/11 = 0.81 10/11 = 0.90
1/12 = 0.083 5/12 = 0.416 7/12 = 0.583 11/12 = 0.916
1/16 = 0.0625 3/16 = 0.1875 5/16 = 0.3125 7/16 = 0.4375
11/16 = 0.6875 13/16 = 0.8125 15/16 = 0.9375
1/32 = 0.03125 3/32 = 0.09375 5/32 = 0.15625 7/32 = 0.21875
9/32 = 0.28125 11/32 = 0.34375 13/32 = 0.40625
15/32 = 0.46875 17/32 = 0.53125 19/32 = 0.59375
21/32 = 0.65625 23/32 = 0.71875 25/32 = 0.78125
27/32 = 0.84375 29/32 = 0.90625 31/32 = 0.96875
Need to convert a repeating decimal to a fraction? Follow these examples:
Note the following pattern for repeating decimals:
0.22222222... = 2/9
0.54545454... = 54/99
0.298298298... = 298/999
Division by 9's causes the repeating pattern.
Note the pattern if zeros precede the repeating decimal:
0.022222222... = 2/90
0.00054545454... = 54/99000
0.00298298298... = 298/99900
Adding zero's to the denominator adds zero's before the repeating decimal.
To convert a decimal that begins with a non-repeating part, such as 0.21456456456456456..., to a fraction, write it as the sum of the non-repeating part and the repeating part.
0.21 + 0.00456456456456456...
Next, convert each of these decimals to fractions. The first decimal has a divisor of power ten. The second decimal (which repeats) is converted according to the pattern given above.
21/100 + 456/99900
Now add these fraction by expressing both with a common divisor
20979/99900 + 456/99900
and add.
21435/99900
Finally simplify it to lowest terms
1429/6660
and check on your calculator or with long division.
= 0.2145645645...
The Compound Interest Equation
P = C (1 + r/n) nt
where
P = future value
C = initial deposit
r = interest rate (expressed as a fraction: eg. 0.06)
n = # of times per year interest is compounded
t = number of years invested
Simplified Compound Interest Equation
When interest is only compounded once per year (n=1), the equation simplifies to:
P = C (1 + r) t
Continuous Compound Interest
When interest is compounded continually (i.e. n --> ), the compound interest equation takes the form:
P = C e rt
Demonstration of Various Compounding
The following table shows the final principal (P), after t = 1 year, of an account initially with C = $10000, at 6% interest rate, with the given compounding (n). As is shown, the method of compounding has little effect. n P
1 (yearly) $ 10600.00
2 (semiannually) $ 10609.00
4 (quarterly) $ 10613.64
12 (monthly) $ 10616.78
52 (weekly) $ 10618.00
365 (daily) $ 10618.31
continuous $ 10618.37
Loan Balance
Situation: A person initially borrows an amount A and in return agrees to make n repayments per year, each of an amount P. While the person is repaying the loan, interest is accumulating at an annual percentage rate of r, and this interest is compounded n times a year (along with each payment). Therefore, the person must continue paying these installments of amount P until the original amount and any accumulated interest is repaid. This equation gives the amount B that the person still needs to repay after t years.
B = A (1 + r/n)NT - P (1 + r/n)NT - 1
--------------------------------------------------------------------------------
(1 + r/n) - 1
where
B = balance after t years
A = amount borrowed
n = number of payments per year
P = amount paid per payment
r = annual percentage rate (APR)
Unit Conversion Tables for Lengths & Distances
A note on the metric system:
Before you use this table, convert to the base measurement first. For example, convert centimeters to meters, convert kilograms to grams.
The notation 1.23E - 4 stands for 1.23 x 10-4 = 0.000123.
from \ to = __ feet = __ inches = __ meters = __ miles = __ yards
foot 12 0.3048 (1/5280) (1/3)
inch (1/12) 0.0254 (1/63360) (1/36)
meter 3.280839... 39.37007... 6.213711...E - 4 1.093613...
mile 5280 63360 1609.344 1760
yard 3 36 0.9144 (1/1760)
To use: Find the unit to convert from in the left column, and multiply it by the expression under the unit to convert to.
Examples: foot = 12 inches; 2 feet = 2x12 inches.
Useful Exact Length Relationships
mile = 1760 yards = 5280 feet
yard = 3 feet = 36 inches
foot = 12 inches
inch = 2.54 centimeters
Unit Conversion Tables for Areas
A note on the metric system:
Before you use this table convert to the base measurement first. For example, convert centimeters to meters, convert kilograms to grams.
The notation 1.23E - 4 stands for 1.23 x 10-4 = 0.000123.
from \ to = __ acres = __ feet2 = __ inches2 = __ meters2 = __ miles2 = __ yards2
acre 43560 6272640 4046.856... (1/640) 4840
foot2 (1/43560) 144 0.09290304 (1/27878400) (1/9)
inch2 (1/6272640) (1/144) 6.4516E - 4 3.587006E - 10 (1/1296)
meter2 2.471054...E - 4 10.76391... 1550.0031 3.861021...E - 7 1.195990...
mile2 640 27878400 2.78784E + 9 2.589988...E + 6 3097600
yard2 (1/4840) 9 1296 0.83612736 3.228305...E - 7
To use: Find the unit to convert from in the left column, and multiply it by the expression under the unit to convert to.
Examples: foot2 = 144 inches2; 2 feet2 = 2x144 inches2.
Useful Exact Area & Length Relationships
acre = (1/640) miles2
mile = 1760 yards = 5280 feet
yard = 3 feet = 36 inches
foot = 12 inches
inch = 2.54 centimeters
Note that when converting area units:
1 foot = 12 inches
(1 foot)2 = (12 inches)2 (square both sides)
1 foot2 = 144 inches2
The linear & area relationships are not the same!
Metric Prefix Table
Number Prefix Symbol
10 1 deka- da
10 2 hecto- h
10 3 kilo- k
10 6 mega- M
10 9 giga- G
10 12 tera- T
10 15 peta- P
10 18 exa- E
10 21 zeta- Z
10 24 yotta- Y
10 -1 deci- d
10 -2 centi- c
10 -3 milli- m
10 -6 micro-
10 -9 nano- n
10 -12 pico- p
10 -15 femto- f
10 -18 atto- a
10 -21 zepto- z
10 -24 yocto- y
Pi
Pi is a name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. Pi is often written using the symbol and is pronounced "pie", just like the dessert.
Brief History of Pi
Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three. The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi.
In 1761 Lambert proved that Pi was irrational, that is, that it can't be written as a ratio of integer numbers.
In 1882 Lindeman proved that Pi was transcendental, that is, that Pi is not the root of any algebraic equation with rational coefficients. This discovery proved that you can't "square a circle", which was a problem that occupied many mathematicians up to that time.How many digits are there? Does it ever end?
Because Pi is known to be an irrational number it means that the digits never end or repeat in any known way. But calculating the digits of Pi has proven to be an fascination for mathematicians throughout history. Some spent their lives calculating the digits of Pi, but until computers, less than 1,000 digits had been calculated. In 1949, a computer calculated 2,000 digits and the race was on. Millions of digits have been calculated, with the record held (as of September 1999) by a supercomputer at the University of Tokyo that calculated 206,158,430,000 digits. (first 1,000 digits)Approximation of Pi
Archimedes calculated that Pi was between 3 10/71 and 3 1/7 (also written 223/71 < < 22/7 ). 22/7 is still a good approximation.
A Cool Pi Experiment
One of the most interesting ways to learn more about Pi is to do pi experiments yourself. Here is a famous one called Buffon's Needle.
In Buffon's Needle experiment you can drop a needle on a lined sheet of paper. If you keep track of how many times the needle lands on a line, it turns out to be directly related to the value of Pi
Vieta's Formula
2/PI = 2/2 * ( 2 + 2 )/2 * (2 + ( ( 2 + 2) ) )/2 * ...c
Leibnitz's Formula
PI/4 = 1/1 - 1/3 + 1/5 - 1/7 + ...
Wallis Product
PI/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...
2/PI = (1 - 1/22)(1 - 1/42)(1 - 1/62)...
Lord Brouncker's Formula
4/PI = 1 + 1
----------------
2 + 32
------------
2 + 52
---------
2 + 72 ...
(PI2)/8 = 1/12 + 1/32 + 1/52 + ...
(PI2)/24 = 1/22 + 1/42 + 1/62 + ...
Euler's Formula
(PI2)/6 = (n = 1..) 1/n2 = 1/12 + 1/22 + 1/32 + ...
(or more generally...)
(n = 1..) 1/n(2k) = (-1)(k-1) PI(2k) 2(2k) B(2k) / ( 2(2k)!)
B(k) = the k th Bernoulli number. eg. B0=1 B1=-1/2 B2=1/6 B4=-1/30 B6=1/42 B8=-1/30 B10=5/66. Further Bernoulli numbers are defined as (n 0)B0 + (n 1)B1 + (n 2)B2 + ... + (n (n-1))B(N-1) = 0 assuming all odd Bernoulli #'s > 1 are = 0. (n k) = binomial coefficient = n!/(k!(n-k)!)
gamma = = 0.5772156649 0153286061 ...
= lim (n->) ( 1 + 1/2 + 1/3 + 1/4 + ... + 1/n - ln(n) ) = 0.5772156649...
= - e^-x ln x dx
Closure Property of Addition
Sum (or difference) of 2 real numbers equals a real number
Additive Identity
a + 0 = a
Additive Inverse
a + (-a) = 0
Associative of Addition
(a + b) + c = a + (b + c)
Commutative of Addition
a + b = b + a
Definition of Subtraction
a - b = a + (-b)
Closure Property of Multiplication
Product (or quotient if denominator 0) of 2 reals equals a real number
Multiplicative Identity
a * 1 = a
Multiplicative Inverse
a * (1/a) = 1 (a 0)
(Multiplication times 0)
a * 0 = 0
Associative of Multiplication
(a * b) * c = a * (b * c)
Commutative of Multiplication
a * b = b * a
Distributive Law
a(b + c) = ab + ac
Definition of Division
a / b = a(1/b)
Conic
By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.
The General Equation for a Conic Section:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
The type of section can be found from the sign of: B2 - 4AC If B2 - 4AC is... then the curve is a...
< 0 ellipse, circle, point or no curve.
= 0 parabola, 2 parallel lines, 1 line or no curve.
> 0 hyperbola or 2 intersecting lines.
The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k).
Circle Ellipse Parabola Hyperbola
Equation (horiz. vertex): x2 + y2 = r2 x2 / a2 + y2 / b2 = 1 4px = y2 x2 / a2 - y2 / b2 = 1
Equations of Asymptotes: y = ± (b/a)x
Equation (vert. vertex): x2 + y2 = r2 y2 / a2 + x2 / b2 = 1 4py = x2 y2 / a2 - x2 / b2 = 1
Equations of Asymptotes: x = ± (b/a)y
Variables: r = circle radius a = major radius (= 1/2 length major axis)
b = minor radius (= 1/2 length minor axis)
c = distance center to focus p = distance from vertex to focus (or directrix) a = 1/2 length major axis
b = 1/2 length minor axis
c = distance center to focus
Eccentricity: 0 c/a 1 c/a
Relation to Focus: p = 0 a2 - b2 = c2 p = p a2 + b2 = c2
Definition: is the locus of all points which meet the condition... distance to the origin is constant sum of distances to each focus is constant distance to focus = distance to directrix difference between distances to each foci is constant
Related Topics: Geometry section on Circles
Polynomial Identities
a+b) 2 = a 2 + 2ab + b 2
(a+b)(c+d) = ac + ad + bc + bd
a 2 - b 2 = (a+b)(a-b) (Difference of squares)
a 3 b 3 = (a b)(a 2 ab + b 2) (Sum and Difference of Cubes)
x 2 + (a+b)x + AB = (x + a)(x + b)
if ax 2 + bx + c = 0 then x = ( -b (b 2 - 4ac) ) / 2a (Quadratic Formula)
Exponential Identities
Powers
x a x b = x (a + b)
x a y a = (xy) a
(x a) b = x (ab)
x (a/b) = bth root of (x a) = ( bth (x) ) a
x (-a) = 1 / x a
x (a - b) = x a / x b
Logarithms
y = logb(x) if and only if x=b y
logb(1) = 0
logb(b) = 1
logb(x*y) = logb(x) + logb(y)
logb(x/y) = logb(x) - logb(y)
logb(x n) = n logb(x)
logb(x) = logb(c) * logc(x) = logc(x) / logc(b)
Algebraic Graphs
Wait for this...
Functions
Synonyms: correspondence, mapping, transformation
Definition: A function is a relation from a domain set to a range set, where each element of the domain set is related to exactly one element of the range set.
An equivalent definition: A function (f) is a relation from a set A to a set B (denoted f: A®B), such that for each element in the domain of A (Dom(A)), the f-relative set of A (f(A)) contains exactly one element.
Trig Functions: Overview
Under its simplest definition, a trigonometric (literally, a "triangle-measuring") function, is one of the many functions that relate one non-right angle of a right triangle to the ratio of the lengths of any two sides of the triangle (or vice versa).
Any trigonometric function (f), therefore, always satisfies either of the following equations:
f(q) = a / b OR f(a / b) = q,
where q is the measure of a certain angle in the triangle, and a and b are the lengths of two specific sides.
This means that
If the former equation holds, we can choose any right triangle, then take the measurement of one of the non-right angles, and when we evaluate the trigonometric function at that angle, the result will be the ratio of the lengths of two of the triangle's sides.
However, if the latter equation holds, we can chose any right triangle, then compute the ratio of the lengths of two specific sides, and when we evaluate the trigonometric function at any that ratio, the result will be measure of one of the triangles non-right angles. (These are called inverse trig functions since they do the inverse, or vice-versa, of the previous trig functions.)
This relationship between an angle and side ratios in a right triangle is one of the most important ideas in trigonometry. Furthermore, trigonometric functions work for any right triangle. Hence -- for a right triangle -- if we take the measurement of one of the triangles non-right angles, we can mathematically deduce the ratio of the lengths of any two of the triangle's sides by trig functions. And if we measure any side ratio, we can mathematically deduce the measure of one of the triangle's non-right angles by inverse trig functions. More importantly, if we know the measurement of one of the triangle's angles, and we then use a trigonometric function to determine the ratio of the lengths of two of the triangle's sides, and we happen to know the lengths of one of these sides in the ratio, we can then algebraically determine the length of the other one of these two sides. (i.e. if we determine that a / b = 2, and we know a = 6, then we deduce that b = 3.)
Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.
Under a certain convention, we label the sides as opposite, adjacent, and hypotenuse relative to our angle of interest q. full explanation
As mentioned previously, the first type of trigonometric function, which relates an angle to a side ratio, always satisfies the following equation:
f(q) = a / b.
Since given any angle q, there are three ways of choosing the numerator (a), and three ways of choosing the denominator (b), we can create the following nine trigonometric functions:
f(q) = opp/opp f(q) = opp/adj f(q) = opp/hyp
f(q) = adj/opp f(q) = adj/adj f(q) = adj/hyp
f(q) = hyp/opp f(q) = hyp/adj f(q) = hyp/hyp
The three diagonal functions shown in red always equal one. They are degenerate and, therefore, are of no use to us. We therefore remove these degenerate functions and assign labels to the remaining six, usually written in the following order:
sine(q) = opp/hyp cosecant(q) = hyp/opp
cosine(q) = adj/hyp secant(q) = hyp/adj
tangent(q) = opp/adj cotangent(q) = adj/opp
Furthermore, the functions are usually abbreviated: sine (sin), cosine (cos), tangent (tan) cosecant (csc), secant (sec), and cotangent (cot).
Do not be overwhelmed. By far, the two most important trig functions to remember are sine and cosine. All the other trig functions of the first kind can be derived from these two functions. For example, the functions on the right are merely the multiplicative inverse of the corresponding function on the left (that makes them much less useful). Furthermore, the sin(x) / COs(x) = (opp/hyp) / (adj/hyp) = opp / adj = tan(x). Therefore, the tangent function is the same as the quotient of the sine and cosine functions (the tangent function is still fairly handy).
sine(q) = opp/hyp CSC(q) = 1/sin(q)
COs(q) = adj/hyp sec(q) = 1/COs(q)
tan(q) = sin(q)/COs(q) cot(q) = 1/tan(q)
Let's examine these functions further. You will notice that there are the sine, secant, and tangent functions, and there are corresponding "co"-functions. They get their odd names from various similar ideas in geometry. You may suggest that the cofunctions should be relabeled to be the multiplicative inverses of the corresponding sine, secant, and tangent functions. However, there is a method to this madness. A cofunction of a given trig function (f) is, by definition, the function obtained after the complement its parameter is taken. Since the complement of any angle q is 90° - q, the the fact that the following relations can be shown to hold:
sine(90° - q) = cosine(q)
secant(90° - q) = cosecant(q)
tangent(90° - q) = cotangent(q)
thus justifying the naming convention.
The trig functions evaluate differently depending on the units on q, such as degrees, radians, or grads. For example, sin(90°) = 1, while sin(90)=0.89399.... explanation
Just as we can define trigonometric functions of the form
f(q) = a / b
that take a non-right angle as its parameter and return the ratio of the lengths of two triangle sides, we can do the reverse: define trig functions of the form
f(a / b) = q
that take the ratio of the lengths of two sides as a parameter and returns the measurement of one of the non-right angles.
Inverse Functions arcsine(opp/hyp) = q arccosecant(hyp/opp) = q
arccosine(adj/hyp) = q arcsecant(hyp/adj) = q
arctangent(opp/adj) = q arccotangent(adj/opp) = q
As before, the functions are usually abbreviated: arcsine (arcsin), arccosine (arccos), arctangent (arctan) arccosecant (arccsc), arcsecant (arcsec), and arccotangent (arccot). According to the standard notation for inverse functions (f-1), you will also often see these written as sin-1, cos-1, tan-1 csc-1, sec-1, and cot-1. Beware: There is another common notation that writes the square of the trig functions, such as (sin(x))2 as sin2(x). This can be confusing, for you then might then be lead to think that sin-1(x) = (sin(x))-1, which is not true. The negative one superscript here is a special notation that denotes inverse functions (not multiplicative inverses).
Trig Functions: Unit Modes
The trig functions evaluate differently depending on the units on q. For example, sin(90°) = 1, while sin(90)=0.89399.... If there is a degree sign after the angle, the trig function evaluates its parameter as a degree measurement. If there is no unit after the angle, the trig function evaluates its parameter as a radian measurement. This is because radian measurements are considered to be the "natural" measurements for angles. (Calculus gives us a justification for this. A partial explanation comes from the formula for the area of a circle sector, which is simplest when the angle is in radians).
Calculator note: Many calculators have degree, radian, and grad modes (360° = 2p rad = 400 grad). It is important to have the calculator in the right mode since that mode setting tells the calculator which units to assume for angles when evaluating any of the trigonometric functions. For example, if the calculator is in degree mode, evaluating sine of 90 results in 1. However, the calculator returns 0.89399... when in radian mode. Having the calculator in the wrong mode is a common mistake for beginners, especially those that are only familiar with degree angle measurements.
For those who wish to reconcile the various trig functions that depend on the units used, we can define the degree symbol (°) to be the value (PI/180). Therefore, sin(90°), for example, is really just an expression for the sine of a radian measurement when the parameter is fully evaluated. As a demonstration, sin(90°) = sin(90(PI/180)) = sin(PI/2). In this way, we only need to tabulate the "natural" radian version of the sine function. (This method is similar to defining percent % = (1/100) in order to relate percents to ratios, such as 50% = 50(1/100) = 1/2.)
Trig: Labeling Sides
Since there are three sides and two non-right angles in a right triangle, the trigonometric functions will need a way of specifying which sides are related to which angle. (It is not-so-useful to know that the ratio of the lengths of two sides equals 2 if we do not know which of the three sides we are talking about. Likewise, if we determine that one of the angles is 40°, it would be nice to know of which angle this statement is true.
We need a way of labeling the sides. Consider a general right triangle:
A right triangle has two non-right angles, and we choose one of these angles to be our angle of interest, which we label "q." ("q" is the Greek letter "theta.")
We can then uniquely label the three sides of the right triangle relative to our choice of q. As the above picture illustrates, our choice of q affects how the three sides get labeled.
We label the three sides in this manner: The side opposite the right angle is called the hypotenuse. This side is labeled the same regardless of our choice of q. The labeling of the remaining two sides depend on our choice of theta; we therefore speak of these other two sides as being adjacent to the angle q or opposite to the angle q. The remaining side that touches the angle q is considered to be the side adjacent to q, and the remaining side that is far away from the angle q is considered to be opposite to the angle q, as shown in the picture.
What is a Polygon?
A closed plane figure made up of several line segments that are joined together. The sides do not cross each other. Exactly two sides meet at every vertex.
Types of Polygons
Regular - all angles are equal and all sides are the same length. Regular polygons are both equiangular and equilateral.
Equiangular - all angles are equal.
Equilateral - all sides are the same length.
Convex - a straight line drawn through a convex polygon crosses at most two sides. Every interior angle is less than 180°.
Concave - you can draw at least one straight line through a concave polygon that crosses more than two sides. At least one interior angle is more than 180°.
Polygon Formulas
(N = # of sides and S = length from center to a corner)
Area of a regular polygon = (1/2) N sin(360°/N) S2
Sum of the interior angles of a polygon = (N - 2) x 180°
The number of diagonals in a polygon = 1/2 N(N-3)
The number of triangles (when you draw all the diagonals from one vertex) in a polygon = (N - 2)
Side - one of the line segments that make up the polygon.
Vertex - point where two sides meet. Two or more of these points are called vertices.
Diagonal - a line connecting two vertices that isn't a side.
Interior Angle - Angle formed by two adjacent sides inside the polygon.
Exterior Angle - Angle formed by two adjacent sides outside the polygon.
Special Polygons
Special Quadrilaterals - square, rhombus, parallelogram, rectangle, and the trapezoid.
Special Triangles - right, equilateral, isosceles, scalene, acute, obtuse.
Polygon Names
Generally accepted names
Sides Name
n N-gon
3 Triangle
4 Quadrilateral
5 Pentagon
6 Hexagon
7 Heptagon
8 Octagon
10 Decagon
12 Dodecagon
Names for other polygons have been proposed.
Sides Name
9 Nonagon, Enneagon 11 Undecagon, Hendecagon
13 Tridecagon, Triskaidecagon 14 Tetradecagon, Tetrakaidecagon
15 Pentadecagon, Pentakaidecagon 16 Hexadecagon, Hexakaidecagon
17 Heptadecagon, Heptakaidecagon 18 Octadecagon, Octakaidecagon
19 Enneadecagon, Enneakaidecagon 20 Icosagon
30 Triacontagon 40 Tetracontagon
50 Pentacontagon 60 Hexacontagon
70 Heptacontagon 80 Octacontagon
90 Enneacontagon 100 Hectogon, Hecatontagon
1,000 Chiliagon 10,000 Myriagon
To construct a name, combine the prefix+suffix
Sides Prefix
20 Icosikai... 30 Triacontakai...
40 Tetracontakai... 50 Pentacontakai...
60 Hexacontakai... 70 Heptacontakai...
80 Octacontakai... 90 Enneacontakai...
+ Sides Suffix +1 ...henagon
+2 ...digon +3 ...trigon
+4 ...tetragon +5 ...pentagon
+6 ...hexagon +7 ...heptagon
+8 ...octagon +9 ...enneagon
Examples:
46 sided polygon - Tetracontakaihexagon
28 sided polygon - Icosikaioctagon
However, many people use the form n-gon, as in 46-gon, or 28-gon instead of these names.
Area Formulas
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements. Examples
square = a 2
rectangle = ab
parallelogram = bh
trapezoid = h/2 (b1 + b2)
circle = pi r 2
ellipse = pi r1 r2
triangle = one half times the base length times the height of the triangle
equilateral triangle =triangle given SAS (two sides and the opposite angle)
= (1/2) a b sin C
triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)
regular polygon = (1/2) n sin(360°/n) S2
when n = # of sides and S = length from center to a corner
Units
Area is measured in "square" units. The area of a figure is the number of squares required to cover it completely, like tiles on a floor.
Area of a square = side times side. Since each side of a square is the same, it can simply be the length of one side squared.
If a square has one side of 4 inches, the area would be 4 inches times 4 inches, or 16 square inches. (Square inches can also be written in2.)
Be sure to use the same units for all measurements. You cannot multiply feet times inches, it doesn't make a square measurement.
The area of a rectangle is the length on the side times the width. If the width is 4 inches and the length is 6 feet, what is the area?
NOT CORRECT .... 4 times 6 = 24
CORRECT.... 4 inches is the same as 1/3 feet. Area is 1/3 feet times 6 feet = 2 square feet. (or 2 sq. ft., or 2 ft2).
Volume Formulas
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a". "b3" means "b cubed", which is the same as "b" times "b" times "b".
Be careful!! Units count. Use the same units for all measurements. Examples
cube = a 3
rectangular prism = a b c
irregular prism = b h
cylinder = b h = pi r 2 h
pyramid = (1/3) b h
cone = (1/3) b h = 1/3 pi r 2 h
sphere = (4/3) pi r 3
ellipsoid = (4/3) pi r1 r2 r3
Units
Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely, like blocks in a box.
Volume of a cube = side times side times side. Since each side of a square is the same, it can simply be the length of one side cubed.
If a square has one side of 4 inches, the volume would be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic inches can also be written in3.)
Be sure to use the same units for all measurements. You cannot multiply feet times inches times yards, it doesn't make a perfectly cubed measurement.
The volume of a rectangular prism is the length on the side times the width times the height. If the width is 4 inches, the length is 1 foot and the height is 3 feet, what is the volume?
NOT CORRECT .... 4 times 1 times 3 = 12
CORRECT.... 4 inches is the same as 1/3 feet. Volume is 1/3 feet times 1 foot times 3 feet = 1 cubic foot (or 1 cu. ft., or 1 ft3).
Surface Area Formulas
In general, the surface area is the sum of all the areas of all the shapes that cover the surface of the object.
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements. Examples
Surface Area of a Cube = 6 a 2
(a is the length of the side of each edge of the cube)
In words, the surface area of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.
Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac
(a, b, and c are the lengths of the 3 sides)
In words, the surface area of a rectangular prism is the are of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.
Surface Area of Any Prism
(b is the shape of the ends)
Surface Area = Lateral area + Area of two ends
(Lateral area) = (perimeter of shape b) * L
Surface Area = (perimeter of shape b) * L+ 2*(Area of shape b)
Surface Area of a Sphere = 4 pi r 2
(r is radius of circle)
Surface Area of a Cylinder = 2 pi r 2 + 2 pi r h
(h is the height of the cylinder, r is the radius of the top)
Surface Area = Areas of top and bottom +Area of the side
Surface Area = 2(Area of top) + (perimeter of top)* height
Surface Area = 2(pi r 2) + (2 pi r)* h
In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle.
You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r2). Since there is both a top and a bottom, that gets multiplied by two.
The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h.
Add those two parts together and you have the formula for the surface area of a cylinder.
Surface Area = 2(pi r 2) + (2 pi r)* h
Tip! Don't forget the units.
These equations will give you correct answers if you keep the units straight. For example - to find the surface area of a cube with sides of 5 inches, the equation is:
Surface Area = 6*(5 inches)2
= 6*(25 square inches)
= 150 sq. inches
Circle
Definition: A circle is the locus of all points equidistant from a central point.
Definitions Related to Circles
arc: a curved line that is part of the circumference of a circle
chord: a line segment within a circle that touches 2 points on the circle.
circumference: the distance around the circle.
diameter: the longest distance from one end of a circle to the other.
origin: the center of the circle
pi (): A number, 3.141592..., equal to (the circumference) / (the diameter) of any circle.
radius: distance from center of circle to any point on it.
sector: is like a slice of pie (a circle wedge).
tangent of circle: a line perpendicular to the radius that touches ONLY one point on the circle.
Diameter = 2 x radius of circle
Circumference of Circle = PI x diameter = 2 PI x radius
where PI = = 3.141592...
Area of Circle: area = PI r2
Length of a Circular Arc: (with central angle )
if the angle is in degrees, then length = x (PI/180) x r
if the angle is in radians, then length = r x
Area of Circle Sector: (with central angle )
if the angle is in degrees, then area = (/360)x PI r2
if the angle is in radians, then area = ((/(2PI))x PI r2
Equation of Circle: (Cartesian coordinates) for a circle with center (j, k) and radius (r): (x-j)^2 + (y-k)^2 = r^2
Equation of Circle:(polar coordinates)for a circle with center(0, 0):r() = radius
for a circle with center with polar coordinates: (c, ) and radius a:
r2 - 2cr cos( - ) + c2 = a2
Equation of a Circle: (parametric coordinates)for a circle with origin (j, k) and radius r: x(t) = r cos(t) + j y(t) = r sin(t) + k
pi = = 3.141592...)
Perimeter Formulas
The perimeter of any polygon is the sum of the lengths of all the sides.
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".Be careful!! Units count. Use the same units for all measurements. Examples
square = 4a
rectangle = 2a + 2b
triangle = a + b + c
circle = 2pi r
circle = pi d (where d is the diameter)
The perimeter of a circle is more commonly known as the circumference.
Units
Be sure to only add similar units. For example, you cannot add inches to feet.
For example, if you need to find the perimeter of a rectangle with sides of 9 inches and 1 foot, you must first change to the same units.
perimeter = 2 ( a + b)
INCORRECT
perimeter = 2(9 + 1) = 2*10 = 20
CORRECT
perimeter = 2( 9 inches + 1 foot)
= 2( 3/4 foot + 1 foot )
= 2(1 3/4 feet)
= 3 1/2 feet
Unproved Theorems
Riemann Hypothesis
zeta(s) = 1/1s + 1/2s + 1/3s + ... (s = a + it) all 0's of zeta(s) in strip 0<=a<=1 lie on central line a=1/2
Twin Primes occur infinitely
Twin primes are primes that are 2 integers apart. Examples include 5 & 7, 17 & 19, 101 & 103
Goldbach's Postulate
Every even # > 2 can be expressed as the sum of 2 primes.
4=2+2, 6=3+3, 8=3+5, 10=5+5, 12=5+7, .. , 100=3+97, ...
Euclid's Parallel Postulate
Through a point, not on a line, there exists exactly 1 line parallel to the given line. (Then there's those non-Euclidean people...)
(k=1..) 1/kn = ?
Although others have found that this expression equals PI2 / 6 when n=2, PI4 / 90 when n = 4 and similar solutions for all possible even values of n, no one has discovered an exact value when n is an odd integer (3, 5, 7, ...) (note: when n=1, the sum does not converge, but it does has relations to the gamma constant).
Vector
Vector Notation: The lower case letters a-h, l-z denote scalars. Uppercase bold A-Z denote vectors. Lowercase bold i, j, k denote unit vectors.
|
|
k
k
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R S= |R| |S| cos ( = angle between them)
R S= S R
(a R) (bS) = (ab) R S
R (S + T)= R S+ R T
R R = |R| 2
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|R x S| = |R| |S| sin ( = angle between both vectors). Direction of R x S is perpendicular to A & B and according to the right hand rule.
| i j k |
R x S = | r1 r2 r3 | = / |r2 r3| |r3 r1| |r1 r2| \
| s1 s2 s3 | \ |s2 s3| , |s3 s1| , |s1 s2| /
S x R = - R x S
(a R) x S = R x (a S) = a (Rx S)
R x (S + T) = R x S + Rx T
R x R = 0
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If a, b, c = angles between the unit vectors i, j,k and R Then the direction cosines are set by:
COs a = (R i) / |R|; COs b = (R j) / |R|; COs c = (R k) / |R|
|R x S| = Area of parrallagram with sides Rand S.
Component of R in the direction of S = |R|COs = (R S) / |S|(scalar result)
Projection of R in the direction of S = |R|COs = (R S) S/ |S| 2 (vector result)
Prelude: A vector, as defined below, is a specific mathematical structure. It has numerous physical and geometric applications, which result mainly from its ability to represent magnitude and direction simultaneously. Wind, for example, had both a speed and a direction and, hence, is conveniently expressed as a vector. The same can be said of moving objects and forces. The location of a points on a cartesian coordinate plane is usually expressed as an ordered pair (x, y), which is a specific example of a vector. Being a vector, (x, y) has a a certain distance (magnitude) from and angle (direction) relative to the origin (0, 0). Vectors are quite useful in simplifying problems from three-dimensional geometry.
Definition:A scalar, generally speaking, is another name for "real number."
Definition: A vector of dimension n is an ordered collection of n elements, which are called components.
Notation: We often represent a vector by some letter, just as we use a letter to denote a scalar (real number) in algebra. In typewritten work, a vector is usually given a bold letter, such as A, to distinguish it from a scalar quantity, such as A. In handwritten work, writing bold letters is difficult, so we typically just place a right-handed arrow over the letter to denote a vector. An n-dimensional vector A has n elements denoted as A1, A2, ..., An. Symbolically, this can be written in multiple ways:
A =
A = (A1, A2, ..., An)
Example: (2,-5), (-1, 0, 2), (4.5), and (PI, a, b, 2/3) are all examples of vectors of dimension 2, 3, 1, and 4 respectively. The first vector has components 2 and -5.
Note: Alternately, an "unordered" collection of n elements {A1, A2, ..., An} is called a "set."
Definition: Two vectors are equal if their corresponding components are equal.
Example: If A = (-2, 1) and B = (-2, 1), then A = B since -2 = -2 and 1 = 1. However, (5, 3) not_equal (3, 5) because even though they have the same components, 3 and 5, the component do not occur in the same order. Contrast this with sets, where {5, 3} = {3, 5}.
Definition: The magnitude of a vector A of dimension n, denoted |A|, is defined as
|A| = sqrt(A1^2 + A2^2 + ... + An^2)
Geometrically speaking, magnitude is synonymous with "length," "distance", or "speed." In the two-dimensional case, the point represented by the vector A = (A1, A2) has a distance from the origin (0, 0) of sqrt(A1^2 + A2^2) according to the pythagorean theorem. In the three-dimension case, the point represented by the vector A = (A1, A2, A3) has a distance from the origin of sqrt(A1^2 + A2^2 + A3^2) according to the three-dimensional form of the Pythagorean theorem (A box with sides a, b, and c has a diagonal of length sqrt(a2+b2+c2) ). With vectors of dimension n greater than three, our geometric intuition fails, but the algebraic definition remains.
Definition: The sum of two vectors A = (A1, A2, ..., An) and B = (B1, B2, ..., Bn) is defined as
A + B = (A1 + B1, A2 + B2, ..., An + Bn)
Note: Addition of vectors is only defined if both vectors have the same dimension.
Example:
(2, -3) + (0, 1) = (2+0, -3+1) = (2, -2).
(0.1, 2) + (-1, PI) = (0.1 + -1, 2 + PI) = (-0.9, 2+PI)
Justification: Physical and geometric applications warrant such a definition. IF a train travels East at 5 meters/second relative to the ground, which will be denoted in vector notation as VT = (0, 5), and a person on the train walks South at 1 meter/second relative to the train, which will be denoted as VP = (-1, 0), THEN the direction and speed that the person is traveling relative to the ground is represented by the vector VG = VT + VP = (0, 5) + (-1, 0) = (0 + -1, 5 + 0) = (-1, 5). This vector has a magnitude of |VG| = sqrt((-1)^2 + 5^2) = sqrt(6) = 2.449..., which means that the person is traveling at about 2.449 meters/second relative to the ground and the net direction is mostly East but slightly South.
Definition: The scalar product of a scalar k by a vector A = (A1, A2, ..., An) is defined as
kA = (kA1, kA2, ..., kAn)
Example:
2(5, -4) = (2*5, 2*-4) = (10, -8)
-3(1, 2) = (-3*1, -3*2) = (-3, -6)
0(3, 1) = (0*3, 0*1) = (0, 0)
1(2, 3) = (1*2, 1*3) = (2, 3)
Note: In general, 0A = (0, 0, ..., 0) and 1A = A, just as in the algebra of scalars. The vector of any dimension n with all zero elements (0, 0, ..., 0) is called the zero vector and is denoted 0.
Complexity
Basic Operations
i = (-1)
i 2 = -1
1 / i = -i
i 4k = 1; i (4k+1) = i; i (4k+2) = -1; i (4k+3) = -i (k = integer)
( i ) = (1/2)+ (1/2) i
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Complex Definitions of Functions and Operations
(a + bi) + (c + di) = (a+c) + (b + d) i
(a + BI) (c + DI) = ac + adi + bci + bdi 2 = (ac - bd) + (ad +bc) i
1/(a + BI) = a/(a 2 + b 2) - b/(a 2 + b 2) i
(a + BI) / (c + DI) = (ac + BD)/(c 2 + d 2) + (BC - ad)/(c 2 +d 2) i
a2 + b2 = (a + BI) (a - BI) (sum of squares)
e (i ) = cos + i sin
n (a + BI) = (COs(b ln n) + i sin(b ln n))n a
if z = r(COs + i sin ) then z n = r n ( COs n+ i sin n )(DeMoivre's Theorem)
if w = r(COs + i sin );n=integer, then there are n complex nth roots (z) of w for k=0,1,..n-1:
z(k) = r (1/n) [ COs( (+ 2(PI)k)/n ) + i sin( (+ 2(PI)k)/n ) ]
if z = r (COs + i sin ) then ln(z) = ln r + i
sin(a + BI) = sin(a)cosh(b) + COs(a)sinh(b) i
COs(a + BI) = COs(a)cosh(b) - sin(a)sinh(b) i
tan(a + BI) = ( tan(a) + i tanh(b) ) / ( 1 - i tan(a) tanh(b))
= ( sech 2(b)tan(a) + sec 2(a)tanh(b) i ) / (1 + tan 2(a)tanh 2(b))
Trigonometric Identities
sin(theta) = a / c
csc(theta) = 1 / sin(theta) = c / a
cos(theta) = b / c
sec(theta) = 1 / cos(theta) = c / b
tan(theta) = sin(theta) / cos(theta) = a / b
cot(theta) = 1/ tan(theta) = b / a
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sin(-x) = -sin(x)
csc(-x) = -csc(x)
cos(-x) = cos(x)
sec(-x) = sec(x)
tan(-x) = -tan(x)
cot(-x) = -cot(x)
sin^2(x) + cos^2(x) = 1
tan^2(x) + 1 = sec^2(x)
cot^2(x) + 1 = csc^2(x)
sin(x y) = sin x cos y cos x sin y
cos(x y) = cos x cosy sin x sin y
tan(x y) = (tan x tan y) / (1 tan x tan y)
sin(2x) = 2 sin x cos x
cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x)
tan(2x) = 2 tan(x) / (1 - tan^2(x))
sin^2(x) = 1/2 - 1/2 cos(2x)
cos^2(x) = 1/2 + 1/2 cos(2x)
sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )
cos x - cos y = -2 sin( (x - y)/2 ) sin( (x + y)/2 )
Trig Table of Common Angles angle 0 30 45 60 90
sin^2(a) 0/4 1/4 2/4 3/4 4/4
cos^2(a) 4/4 3/4 2/4 1/4 0/4
tan^2(a) 0/4 1/3 2/2 3/1 4/0
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Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C:
a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)
c^2 = a^2 + b^2 - 2ab cos(C)
b^2 = a^2 + c^2 - 2ac cos(B) (Law of Cosines)
a^2 = b^2 + c^2 - 2bc cos(A)
(a - b)/(a + b) = tan [(A-B)/2] / tan [(A+B)/2] (Law of Tangents)
PI = 3.141592... (approximately 22/7 = 3.1428)
radians = degrees x PI / 180 (deg to rad conversion)
degrees = radians x 180 / PI (rad to deg conversion)
Rad Deg Sin Cos Tan Csc Sec Cot
.0000 00 .0000 1.0000 .0000 ----- 1.0000 ----- 90 1.5707
.0175 01 .0175 .9998 .0175 57.2987 1.0002 57.2900 89 1.5533
.0349 02 .0349 .9994 .0349 28.6537 1.0006 28.6363 88 1.5359
.0524 03 .0523 .9986 .0524 19.1073 1.0014 19.0811 87 1.5184
.0698 04 .0698 .9976 .0699 14.3356 1.0024 14.3007 86 1.5010
.0873 05 .0872 .9962 .0875 11.4737 1.0038 11.4301 85 1.4835
.1047 06 .1045 .9945 .1051 9.5668 1.0055 9.5144 84 1.4661
.1222 07 .1219 .9925 .1228 8.2055 1.0075 8.1443 83 1.4486
.1396 08 .1392 .9903 .1405 7.1853 1.0098 7.1154 82 1.4312
.1571 09 .1564 .9877 .1584 6.3925 1.0125 6.3138 81 1.4137
.1745 10 .1736 .9848 .1763 5.7588 1.0154 5.6713 80 1.3953
.1920 11 .1908 .9816 .1944 5.2408 1.0187 5.1446 79 1.3788
.2094 12 .2079 .9781 .2126 4.8097 1.0223 4.7046 78 1.3614
.2269 13 .2250 .9744 .2309 4.4454 1.0263 4.3315 77 1.3439
.2443 14 .2419 .9703 .2493 4.1336 1.0306 4.0108 76 1.3265
.2618 15 .2588 .9659 .2679 3.8637 1.0353 3.7321 75 1.3090
.2793 16 .2756 .9613 .2867 3.6280 1.0403 3.4874 74 1.2915
.2967 17 .2924 .9563 .3057 3.4203 1.0457 3.2709 73 1.2741
.3142 18 .3090 .9511 .3249 3.2361 1.0515 3.0777 72 1.2566
.3316 19 .3256 .9455 .3443 3.0716 1.0576 2.9042 71 1.2392
.3491 20 .3420 .9397 .3640 2.9238 1.0642 2.7475 70 1.2217
.3665 21 .3584 .9336 .3839 2.7904 1.0711 2.6051 69 1.2043
.3840 22 .3746 .9272 .4040 2.6695 1.0785 2.4751 68 1.1868
.4014 23 .3907 .9205 .4245 2.5593 1.0864 2.3559 67 1.1694
.4189 24 .4067 .9135 .4452 2.4586 1.0946 2.2460 66 1.1519
.4363 25 .4226 .9063 .4663 2.3662 1.1034 2.1445 65 1.1345
.4538 26 .4384 .8988 .4877 2.2812 1.1126 2.0503 64 1.1170
.4712 27 .4540 .8910 .5095 2.2027 1.1223 1.9626 63 1.0996
.4887 28 .4695 .8829 .5317 2.1301 1.1326 1.8807 62 1.0821
.5061 29 .4848 .8746 .5543 2.0627 1.1434 1.8040 61 1.0647
.5236 30 .5000 .8660 .5774 2.0000 1.1547 1.7321 60 1.0472
.5411 31 .5150 .8572 .6009 1.9416 1.1666 1.6643 59 1.0297
.5585 32 .5299 .8480 .6249 1.8871 1.1792 1.6003 58 1.0123
.5760 33 .5446 .8387 .6494 1.8361 1.1924 1.5399 57 .9948
.5934 34 .5592 .8290 .6745 1.7883 1.2062 1.4826 56 .9774
.6109 35 .5736 .8192 .7002 1.7434 1.2208 1.4281 55 .9599
.6283 36 .5878 .8090 .7265 1.7013 1.2361 1.3764 54 .9425
.6458 37 .6018 .7986 .7536 1.6616 1.2521 1.3270 53 .9250
.6632 38 .6157 .7880 .7813 1.6243 1.2690 1.2799 52 .9076
.6807 39 .6293 .7771 .8098 1.5890 1.2868 1.2349 51 .8901
.6981 40 .6428 .7660 .8391 1.5557 1.3054 1.1918 50 .8727
.7156 41 .6561 .7547 .8693 1.5243 1.3250 1.1504 49 .8552
.7330 42 .6691 .7431 .9004 1.4945 1.3456 1.1106 48 .8378
.7505 43 .6820 .7314 .9325 1.4663 1.3673 1.0724 47 .8203
.7679 44 .6947 .7193 .9657 1.4396 1.3902 1.0355 46 .8029
.7854 45 .7071 .7071 1.0000 1.4142 1.4142 1.0000 45 .7854
COs Sin Cot Sec CSC Tan Deg Rad
Trig Table of Common Angles angle (degrees) 0 30 45 60 90 120 135 150 180 210 225 240 270 300 315 330 360 = 0
angle (radians) 0 PI/6 PI/4 PI/3 PI/2 2/3PI 3/4PI 5/6PI PI 7/6PI 5/4PI 4/3PI 3/2PI 5/3PI 7/4PI 11/6PI 2PI = 0
sin(a) (0/4) (1/4) (2/4) (3/4) (4/4) (3/4) (2/4) (1/4) (0/4) -(1/4) -(2/4) -(3/4) -(4/4) -(3/4) -(2/4) -(1/4) (0/4)
COs(a) (4/4) (3/4) (2/4) (1/4) (0/4) -(1/4) -(2/4) -(3/4) -(4/4) -(3/4) -(2/4) -(1/4) (0/4) (1/4) (2/4) (3/4) (4/4)
tan(a) (0/4) (1/3) (2/2) (3/1) (4/0) -(3/1) -(2/2) -(1/3) -(0/4) (1/3) (2/2) (3/1) (4/0) -(3/1) -(2/2) -(1/3) (0/4)
Those with a zero in the denominator are undefined. They are included solely to demonstrate the
Hyperbolic Definitions
sinh(x) = ( e x - e -x )/2
csch(x) = 1/sinh(x) = 2/( e x - e -x )
cosh(x) = ( e x + e -x )/2
sech(x) = 1/cosh(x) = 2/( e x + e -x )
tanh(x) = sinh(x)/cosh(x) = ( e x - e -x )/( e x + e -x )
coth(x) = 1/tanh(x) = ( e x + e -x)/( e x - e -x )
cosh 2(x) - sinh 2(x) = 1
tanh 2(x) + sech 2(x) = 1
coth 2(x) - csch 2(x) = 1
Inverse Hyperbolic Definitions
arcsinh(z) = ln( z + (z 2 + 1) )
arccosh(z) = ln( z (z 2 - 1) )
arctanh(z) = 1/2 ln( (1+z)/(1-z) )
arccsch(z) = ln( (1+(1+z 2) )/z )
arcsech(z) = ln( (1(1-z 2) )/z )
arccoth(z) = 1/2 ln( (z+1)/(z-1) )
Relations to Trigonometric Functions
sinh(z) = -i sin(iz)
csch(z) = i csc(iz)
cosh(z) = cos(iz)
sech(z) = sec(iz)
tanh(z) = -i tan(iz)
coth(z) = i cot(iz)
Definitions of the Derivative:
df / dx = lim (dx -> 0) (f(x+dx) - f(x)) / dx (right sided)
df / dx = lim (dx -> 0) (f(x) - f(x-dx)) / dx (left sided)
df / dx = lim (dx -> 0) (f(x+dx) - f(x-dx)) / (2dx) (both sided)
f(t) dt = f(x) (Fundamental Theorem for Derivatives)
--------------------------------------------------------------------------------
c f(x) = c
f(x) (c is a constant)
(f(x) + g(x)) = f(x) + g(x)
f(g(x)) = f(g) * g(x) (chain rule)
f(x)g(x) = f'(x)g(x) + f(x)g '(x) (product rule)
f(x)/g(x) = ( f '(x)g(x) - f(x)g '(x) ) / g^2(x) (quotient rule)
--------------------------------------------------------------------------------
Partial Differentiation Identities
if f( x(r,s), y(r,s) )
df / dr = df / dx * dx / DR + df / dy * dy / DR
df / ds = df / dx * dx / Ds + df / dy * dy / Ds
if f( x(r,s) )
df / DR = df / dx * dx / DR
df / Ds = df / dx * dx / Ds
directional derivative
df(x,y) / d(Xi sub a) = f1(x,y) cos(a) + f2(x,y) sin(a)
(Xi sub a) = angle counterclockwise from pos. x axis.
Derivatives: Min, Max, Critical Points
Asymptotes
Definition of a horizontal asymptote: The line y = y0 is a "horizontal asymptote" of f(x) if and only if f(x) approaches y0 as x approaches + or - .
Definition of a vertical asymptote: The line x = x0 is a "vertical asymptote" of f(x) if and only if f(x) approaches + or - as x approaches x0 from the left or from the right.
Definition of a slant asymptote: the line y = ax + b is a "slant asymptote" of f(x) if and only if lim (x-->+/-) f(x) = ax + b.
Concavity
Definition of a concave up curve: f(x) is "concave up" at x0 if and only if f '(x) is increasing at x0
Definition of a concave down curve: f(x) is "concave down" at x0 if and only if f '(x) is decreasing at x0
The second derivative test: If f ''(x) exists at x0 and is positive, then f ''(x) is concave up at x0. If f ''(x0) exists and is negative, then f(x) is concave down at x0. If f ''(x) does not exist or is zero, then the test fails.
Critical Points
Definition of a critical point: a critical point on f(x) occurs at x0 if and only if either f '(x0) is zero or the derivative doesn't exist.
Extrema (Maxima and Minima)
Local (Relative) Extrema
Definition of a local maxima: A function f(x) has a local maximum at x0 if and only if there exists some interval I containing x0 such that f(x0) >= f(x) for all x in I.
Definition of a local minima: A function f(x) has a local minimum at x0 if and only if there exists some interval I containing x0 such that f(x0) <= f(x) for all x in I.
Occurrence of local extrema: All local extrema occur at critical points, but not all critical points occur at local extrema.
The first derivative test for local extrema: If f(x) is increasing (f '(x) > 0) for all x in some interval (a, x0] and f(x) is decreasing (f '(x) < 0) for all x in some interval [x0, b), then f(x) has a local maximum at x0. If f(x) is decreasing (f '(x) < 0) for all x in some interval (a, x0] and f(x) is increasing (f '(x) > 0) for all x in some interval [x0, b), then f(x) has a local minimum at x0.
The second derivative test for local extrema: If f '(x0) = 0 and f ''(x0) > 0, then f(x) has a local minimum at x0. If f '(x0) = 0 and f ''(x0) < 0, then f(x) has a local maximum at x0.
Absolute Extrema
Definition of absolute maxima: y0 is the "absolute maximum" of f(x) on I if and only if y0 >= f(x) for all x on I.
Definition of absolute minima: y0 is the "absolute minimum" of f(x) on I if and only if y0 <= f(x) for all x on I.
The extreme value theorem: If f(x) is continuous in a closed interval I, then f(x) has at least one absolute maximum and one absolute minimum in I.
Occurrence of absolute maxima: If f(x) is continuous in a closed interval I, then the absolute maximum of f(x) in I is the maximum value of f(x) on all local maxima and endpoints on I.
Occurrence of absolute minima: If f(x) is continuous in a closed interval I, then the absolute minimum of f(x) in I is the minimum value of f(x) on all local minima and endpoints on I.
Alternate method of finding extrema: If f(x) is continuous in a closed interval I, then the absolute extrema of f(x) in I occur at the critical points and/or at the endpoints of I.
(This is a less specific form of the above.)
Increasing/Decreasing Functions
Definition of an increasing function: A function f(x) is "increasing" at a point x0 if and only if there exists some interval I containing x0 such that f(x0) > f(x) for all x in I to the left of x0 and f(x0) < f(x) for all x in I to the right of x0.
Definition of a decreasing function: A function f(x) is "decreasing" at a point x0 if and only if there exists some interval I containing x0 such that f(x0) < f(x) for all x in I to the left of x0 and f(x0) > f(x) for all x in I to the right of x0.
The first derivative test: If f '(x0) exists and is positive, then f '(x) is increasing at x0. If f '(x) exists and is negative, then f(x) is decreasing at x0. If f '(x0) does not exist or is zero, then the test tells fails.
Inflection Points
Definition of an inflection point: An inflection point occurs on f(x) at x0 if and only if f(x) has a tangent line at x0 and there exists and interval I containing x0 such that f(x) is concave up on one side of x0 and concave down on the other side.
The fourier series of the function f(x)
a(0) / 2 + (k=1..) (a(k) cos kx + b(k) sin kx)
a(k) = 1/PI f(x) cos kx dx
b(k) = 1/PI f(x) sin kx dx
Remainder of fourier series. Sn(x) = sum of first n+1 terms at x.
remainder(n) = f(x) - Sn(x) = 1/PI f(x+t) Dn(t) dt
Sn(x) = 1/PI f(x+t) Dn(t) dt
Dn(x) = Dirichlet kernel = 1/2 + cos x + cos 2x + .. + cos nx = [ sin(n + 1/2)x ] / [ 2sin(x/2) ]
Riemann's Theorem. If f(x) is continuous except for a finite # of finite jumps in every finite interval then:
lim(k->) f(t) cos kt dt = lim(k->)f(t) sin kt dt = 0
The fourier series of the function f(x) in an arbitrary interval.
A(0) / 2 + (k=1..) [ A(k) cos (k(PI)x / m) + B(k) (sin k(PI)x / m) ]
a(k) = 1/m f(x) cos (k(PI)x / m) dx
b(k) = 1/m f(x) sin (k(PI)x / m) dx
Parseval's Theorem. If f(x) is continuous; f(-PI) = f(PI) then
1/PI f^2(x) dx = a(0)^2 / 2 + (k=1..) (a(k)^2 + b(k)^2)
Fourier Integral of the function f(x)
f(x) = ( a(y) cos yx + b(y) sin yx ) dy
a(y) = 1/PI f(t) cos ty dt
b(y) = 1/PI f(t) sin ty dt
f(x) = 1/PI dy f(t) cos (y(x-t)) dt
Special Cases of Fourier Integral
if f(x) = f(-x) then
f(x) = 2/PI cos xy dy f(t) cos yt dt
if f(-x) = -f(x) then
f(x) = 2/PI sin xy dy sin yt dt
Fourier Transforms
Fourier Cosine Transform
g(x) = (2/PI)f(t) cos xt dt
Fourier Sine Transform
g(x) = (2/PI)f(t) sin xt dt
Identities of the Transforms
If f(-x) = f(x) then
Fourier Cosine Transform ( Fourier Cosine Transform (f(x)) ) = f(x)
If f(-x) = -f(x) then
Fourier Sine Transform (Fourier Sine Transform (f(x)) ) = f(x)
Recursive Formulas for y1/n
Explicit form:
y1/n = x
Recursive form:
xk+1 = (xk + y / (xk)n-1) / 2
where y ³ 0 and n > 0
or y Î Â and n is odd, positive, and integer.
(for negative n, evaluate the above formula with n positive, then invert your answer).
Example: 21/3 = 1.259921049894...
iteration value
x0 1.0000000000
x1 1.5000000000
x2 1.1944444444
x3 1.2981416381
x4 1.2424821566
x5 1.2690093603
x6 1.2554742937
x7 1.2621680807
x8 1.2588035314
x9 1.2604812976
x10 1.2596412994
x20 1.2599207765
x50 1.259921049894
x 1.259921049894...
Source: Jeff Yates, et al.
Recursive Formulas for B/A
Explicit form:
Find B/A where B and A are real numbers and B > 0
Recursive form:
Convert B and A to scientific notation base 2 (C++ has function "frexp" for this). Note: mantissa is ³ .5 and < 1.
Let
a = mantissa of A
b = mantissa of B
exp = exponent of b - exponent of a
x0 = 1
xn+1 = xn(2 - a xn)
Reiterate x until desired precision reached. Result only has to be close, not perfect. I suggest about 5 times.
y0 = b xn
yi+1 = yi + xn(b - a yi)
Reiterate y until desired precision reached.
B / A = yi * 2exp
Example: 314.51 / 5.6789
Written in base-2 scientific notation:
B = 0.61357421875 * 29
A = 0.7098625 * 23
exp = 9 - 3 = 6
iteration value
x0 1
x1 1.2901375
x2 1.398740976607287
x3 1.408652781763906
x4 1.408723516847958
(will use this value for x)
iteration value
y0 0.864356431284738
y1 0.864356433463227
y2 0.864356433463227
y3 0.864356433463227
y4 0.864356433463227
y5 0.864356433463227
B / A = 0.864356433463227 * 26 =
55.318811741710538
55.318811741710538 (true value)
Recursive Formulas for (A)
Explicit form:
Find (A), where A is a real number > 0.
Recursive form:
Convert A to scientific notation base 2 (C++ has function "frexp" for this). Note: mantissa is ³ .5 and < 1.
Let
a = mantissa of A
exp = exponent of a
a = a * 2(exp mod 2)
exp = exp \ 2 (Note: integer divide)
xn+1 = (xn/2)(3 - axn2)
Reiterate about 5 or 6 times then do y with that result.
yi+1 = yi + (xn/2)(a - yi2)
Reiterate until required precision attained.
A = yi * 2exp
Formula Derivations - (High School +) Derivations of area, perimeter, volume and more for 2 and 3 dimensional figures. (Math Forum)
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