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).