Math Review

PEP 294: Lecture Notes

Trigonometry & Vector Algebra

Trigonometry (pp. 504-507)

1. Basic Geometry

Lines & angles:

- straight line:

a = 180°

- parallel lines:

a = b = c = d

Triangles:

- regular triangle:

3 sides & 3 angles
a + b + c = 180°

- right-angled triangle:

c = 90°
a + b + c = 180°
a + b = 90° => b = a - 90°

C = hypotenuse
A = opposite of a (or adjacent of b)
B = adjacent of a (or opposite of b)

Pythagorean theorem:

- Example: A = 3 & B = 4. Compute C.

2. Trigonometric Functions

Definitions:

SOH CAH TOA!

Properties:

=>

Example 1: For the right triangle above, A = 3 & B = 4. Compute the following.

- From the Pythagorean theorem:

C =

- Therefore:

cos a =

sin a =

tan a =

cos b =

sin b =

tan b =

(Use 3 decimal places for the trig function values.)

Example 2 (use a calculator)

sin 30° =

cos 60° =

tan 60° =
(Use 3 decimal places for the trig function values.)

Inverse functions:

- known function value => compute angle

sin a = x => a = sin^-1(x), -90° < a < 90°

cos a = x => a = cos^-1(x), 0° < a < 180°

tan a = x => a = tan^-1(x), -90° < a < 90°

- Examples:

sin a = 0.643 => a = sin^-1( 0.643 ) =

cos a = 0.921 => a =

tan a = 1.483 => a =

Vector Algebra (pp. 76-81)

1. Vector vs. Scalar Quantity

Displacement vs. distance:

- moving from A to B (Path 1 or 2)

- distance:

(a) length of the path (magnitude)

(b) varies depending on the path (1 or 2)

(c) scalar: magnitude only

- displacement:

(a) length of straight path from A to B (magnitude) + NE (direction)

(b) net effect of motion

(c) path-independent (1 = 2)

(d) vector: magnitude + direction

Vector

- magnitude + direction

- examples: displacement, position, velocity, force, momentum….

- a (bold) or (arrow)

- vector arrow:

(a) length of the arrow: magnitude

(b) tip: direction

Vector example: force

- differences in magnitude and direction of the force result in different motions of the ball

2. Vector Algebra

Vector addition:

- displacement vectors

The "Tip-To-Tale Method": the graphical approach

- connect all vectors, one by one, tip to tail

- move vectors freely while keeping direction and magnitude

- sequence of addition is not important

Vector resolution:

- resolution: breaking down of a vector into components

a => a_X + a_Y

- a_X & a_Y: components:

- Example: Soccer kick, velocity = 20 m/s, q = 35° --> v_H, v_V ?

v_H = v·cosq = (20)(cos 35°) = 16.38 m/s

v_V = v·sinq = (20)(sin 35°) = 11.47 m/s

Vector composition:

- composition: construction of a vector from its components

a_X + a_Y => a

- Example (Sample problem 2 on p. 302): v_swim = 2 m/s, v_current = 0.5 m/s --> v, q = ?

v = = 1.62 m/s

q = tan^-1 = 14.04°

Vector addition: the component approach

- Let C = A + B, then

C_x = A_x + B_x

C_y = A_y + B_y

- Sequences:

(a) Compute the components of A and B through vector resolution. (Pat attention to the directions of the components.)

(b) Compute the components of C (C_x and C_y).

(c) Compute the magnitude (C) and the direction (q) of C through vector composition.

- Example:

Displacements of an orienteer: 400m E + 500 m NE

A_x = 400, A_y = 0

B_x = (500)(cos45°) = 353.50

B_y = (500)(sin45°) = 353.50

C_x = A_x + B_x = 400 + 353.50 = 753.50

C_y = A_y + B_y = 0 + 353.50 = 353.50

C =

q =

Ans: 832.30 m (25.13° N of due E)