# Math Review

**I. This is calculus-based physics
A. but good algebra skills are very important
1. We’ll actually do a lot more algebra than calculus
B. You’ll also need to remember some geometry and trigonometry
C. So, we’ll have a quick math review …
II. Algebra**

**A. Manipulating equations**

1. Do algebra, and save the number-crunching for last

a) minimize button-pushing errors on calculator

b) easier to track units on numbers and make sure the units come out correctly (more later)

2. Example: solve this equation for “a”

a)

1. Do algebra, and save the number-crunching for last

a) minimize button-pushing errors on calculator

b) easier to track units on numbers and make sure the units come out correctly (more later)

2. Example: solve this equation for “a”

a)

**B. Solving simultaneous equations
**

**1. Example: what are x and y in**4x-y=5 and 3x+2y=12 ?

**a) Method 1:
**

**b) Method 2:**

**C. Solving quadratic equations
1. Quadratic equation: variable raised to power of 2
a) **x

^{2}-x-6=0

**2. Factoring a quadratic equation**

**a) Put the equation in the form **

(x +a) (x +b)=0

(1) or x^{2}+ax+bx+ab=0 ⇒x^{2}+(a+b)x+ab=0

(2) We need the factors of -6 such that

**(a) they equal -6 when multiplied together
(b) and they equal -1 when added together**

(3) Factors are: (-1,6), (1,-6), (2,-3), (3,-2)

**(a) only (2,-3) satisfies both requirements**

b) So the result is (x+2)(x-3)=0

(1) and the possible values for x are

x +2=0⇒x=-2

x-3=0⇒x=3

**3. Using the quadratic formula**

**a) The quadratic formula is a general solution to the quadratic equation**

ax^{2}+bx+c=0

(1) quadratic formula is

(2) Two answers (due to ±) – which is the one we want?

**(a) In physics, one answer will often “make sense” more than other
(b) The correct choice may depend on the question being asked**

**b) Example:** 3x^{2}+5x-2=0

**III. Geometry**

**A. Areas and volumes of geometric shapes**

**
1. Rectangle (square): **A=l*w, P=2l+2w

**2. Triangle:**

**3. Circle: **P=π*d=2πr,

**4. Box (cube)**: V=l*w*h, A=sum of areas of faces

**5. Sphere:**

**6. Cylinder: A** _{
lateral}**
=**2πr*h,** A**
_{total=}** A** _{
lateral}+2(πr^{2}), V=πr^{2}h

**IV. Trigonometry**

**A. Degrees and radians
1. units for measuring angles**

**a) Make sure you have your calculator set in the
correct mode!
2. Moving in a complete circle goes through 360° or 2π radians
a) hence, to convert between them:
(1) degrees to radians:
(2) radians to degrees:**

**3. Right angles are 90° (π/2 rad)
4. The three angles in a triangle sum to 180° (π rad)
**

**B. Measuring angles on a Cartesian plot**

1. We’ll frequently specify the direction of a line , relative to the Cartesian (x-y) axes

2. The most common method is measuring CCW from the +x axis

a) e.g., illustrate 30°, 45°, 90°, and 180°

b) measuring CW is the negative direction: -60° (300°), -120° (240°)

3. Another method is relative to cardinal directions (N,S,E,W)

a) 20° W of N, 30° S of E, due south, etc.

C. Trig functions

1. Sine, cosine, and tangent

a) Defined relative to the sides of a right triangle

1. We’ll frequently specify the direction of a line , relative to the Cartesian (x-y) axes

2. The most common method is measuring CCW from the +x axis

a) e.g., illustrate 30°, 45°, 90°, and 180°

b) measuring CW is the negative direction: -60° (300°), -120° (240°)

3. Another method is relative to cardinal directions (N,S,E,W)

a) 20° W of N, 30° S of E, due south, etc.

C. Trig functions

1. Sine, cosine, and tangent

a) Defined relative to the sides of a right triangle

(1) two “regular” sides and the hypotenuse (opp. the right
angle)

(2) Sine: opposite side over the hypotenuse

(3) Cosine: adjacent side over the hypotenuse

(4) Tangent: opposite side over adjacent side

(5) Useful mnemonic: Soh-Cah-Toa

**b) Useful trig identities (consult a math handbook!)
**

**2. Some special right triangles**

a) 1,2, right triangle** (a 30° angle and a
60° angle)**

b) 1,1, right triangle** (two 45° angles)**

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