Trig

Angle Basics

Positive angles are measured counterclockwise, and negative angles are measured clockwise. Standard position has the vertex at the origin and the initial side on the positive x-axis.

Coterminal Angles

Angles that differ by full rotations, expressed as $\theta \pm 360\degree$ (or $\theta \pm 2\pi$).

Quadrantal Angles

Angles that are multiples of $90\degree$ ($\pi/2$).

Similar Triangles

Triangles that have the same angles will have equal side ratios, which justifies trigonometric ratios.

Sine Definition

For an acute angle $\theta$ in a right triangle, $\sin \theta = \frac{\text{opp}}{\text{hyp}}$.

Cosine Definition

For an acute angle $\theta$ in a right triangle, $\cos \theta = \frac{\text{adj}}{\text{hyp}}$.

Tangent Definition

For an acute angle $\theta$ in a right triangle, $\tan \theta = \frac{\text{opp}}{\text{adj}}$.

Reciprocal Trig Functions

The reciprocals are defined as: $\csc = \frac{1}{\sin}, \sec = \frac{1}{\cos}, \cot = \frac{1}{\tan}$.

Pythagorean Identity

$\sin^2\theta + \cos^2\theta = 1$.

Quadrant Sign Rules

ASTC: QI (All Positive), QII (Sin Positive), QIII (Tan Positive), QIV (Cos Positive).

Cofunction Identity

For acute angles, $\sin \theta = \cos(90^{\circ} - \theta)$ and vice versa.

Special Angles

The values for $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$ should be memorized for $\sin, \cos, \tan$.

Arc Length Formula

Arc length is given by $s = r \theta$, where $\theta$ is in radians.

Sector Area Formula

The area of a sector is given by $A = \frac{1}{2} r^2 \theta$.

Unit Circle Basics

On the unit circle, $\sin \theta = y$, $\cos \theta = x$, and $\tan \theta = \frac{y}{x}$.

Angular Speed

Angular speed is defined as $\omega = \frac{\theta}{t}$ (in rad/s).

Linear Speed

Linear speed is given by the formula $v = r \omega$.

Periodicity of Functions

The period for $\sin$ and $\cos$ is $2\pi$, while it is $\pi$ for $\tan$ and $\cot$.

Transformation of Graphs

The transformation of sine and cosine functions can be expressed as $y = A \sin(B(x - C)) + D$.

Harmonic Motion Representation

Harmonic motion can be modeled as $x(t) = A \cos(\omega t + \phi)$ or $A \sin(\omega t + \phi)$.