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Math Review

Objectives:

• Review basic concepts in algebra, geometry, trigonometry, and statistics

Negative Numbers

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Indicate positions and directions opposite to the defined positive direction

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change of +3 from 1:

change of –3 from 1:

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Operations with Negative Numbers

Addition:Adding a negative number

Subtracting a positive number

Subtraction:Subtracting a negative number

Adding a positive number

Multiplication & Division:Numbers have same sign Result is positive

Numbers have opposite signs Result is negative

Exponents & Square Roots

Exponent: indicates repeated self-multiplication

52 = 5 × 5 = 25

53 = 5 × 5 × 5 = 125

Square Root ( ): inverse of squaring a number

x2 = x

52 = 25 25 = 52 = 5

Alternate notation: x = x1/2

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Order of Operations

When computing the value of an expression, operations are performed in the following order (or precedence):

1. Expressions in parentheses

2. Exponents (N or ^) and square roots ( )

3. Multiplication (× or *) and division (/)

4. Addition (+) and subtraction (–)

Multiple operations with the same precedence are performed from left to right.

Simple AlgebraIf:

(expression1) = (expression2)

Then: (expression1) + N = (expression2) + N

(expression1) – N = (expression2) – N

(expression1) * N = (expression2) * N

(expression1) / N = (expression2) / N

(expression1)N = (expression2)N

where N is any number, variable, or expression.

Use these operations to isolate the unknown variable on one side of the equals sign.

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Coordinate Systems

(0,0) x (m)

y (m)

2

2

A coordinate system can be used to quantify position and/or direction

1

1

(x,y) = (1m, 2m)

units of measure

origin

coordinate axes (oriented at 90° to each other)

θ

Lines

(0,0)x

yEquation of a line:

y = m x + b

b : y-interceptm : slope

b

∆x

∆y

m =∆y∆x

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Quadratic Equations

Quadratic equation:

ax2 + bx + c = 0

A quadratic equation has two solutions:

x = 2a

– b + b2 – 4ac

x = 2a

– b – b2 – 4ac

Measuring Angles

Note: Excel uses radians!

0, 2π

π/2

π

3π/2

π = 3.14159

0, 360

90

180

270

Degrees: Radians:

θ(degrees) = (180/π)× θ(radians)

θ(radians) = (π/180)× θ(degrees)

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Geometric Relationships

α

β

γα

α + β + γ = 180°

α

α

α

180°– α

For any triangle:

For parallel & intersecting lines:

parallel

parallel

Pythagorean Theorem

a

bc

c2 = a2 + b2

For a right triangle:

90°

c = a2 + b2

a = c2 – b2

b = c2 – a2

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Trigonometric Functions

a

oh

θ

o : side opposite to θa : side adjacent to θh : hypotenuse

sin θ = oh

sine:

cos θ = ah

cosine:

tan θ = oa

tangent:

90°

cos θ = ah

cosine:

Inverse Trigonometric Functions

a

oh

θ

θ = asin oh

arcsine:

)(

θ = acos ah

arccosine:

)(

θ = atan oa

arctangent:

)(

sin θ = oh

sine:

tan θ = oa

tangent:90°

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Law of Sines

= sin α

asin β

b=

sin γc

a

b

c

α

β

γ

Law of Cosines

a2 = b2 + c2 – 2bc cos α

b2 = a2 + c2 – 2ac cos β

c2 = a2 + b2 – 2ab cos γ

a

b

c

α

β

γ

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Percentages

1% = = 0.01

% change = × 100%

1100

(new value) – (old value)(old value)

What is 45% of 5? 45% × 5 = × 5 = 2.25

2 is what % of 5? 25

× 100% = 0.4 × 100% = 40%

45100

Subscripts & Summations

Variables with similar meaning are often given the same name, but with a different subscript:

x i = x j + x j+1 + x j+2 + ··· + x kΣi = j

k

x i = x3 + x4 + x5Σi = 3

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Summation symbol indicates addition of a sequence of subscripted variables:

e.g. v1 , v2 , vx , vy , vknee , vhip

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Average (or Mean)

The “expected” value of a group of numbers

x iΣi = 1

N

Nx =

add values together, then divide by the number of values

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average

Standard DeviationMeasure of the scatter about the average value

σx =(x i – x )2Σ

i = 1

N

N – 1

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σσx = 3.7

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σσx = 1.8average