Trigonometry Identities and Equations

A set of vocabulary flashcards covering core trigonometric functions, identities, and solution concepts presented in the lecture notes.

Sine (sin x)

Ratio of the length of the opposite side to the hypotenuse in a right triangle; y-coordinate on the unit circle.

Cosine (cos x)

Ratio of the length of the adjacent side to the hypotenuse; x-coordinate on the unit circle.

Tangent (tan x)

Ratio sin x ⁄ cos x; slope of the line from the origin to the point (cos x, sin x) on the unit circle.

Cotangent (cot x)

Reciprocal of tangent; cos x ⁄ sin x.

Secant (sec x)

Reciprocal of cosine; 1 ⁄ cos x.

Cosecant (csc x)

Reciprocal of sine; 1 ⁄ sin x.

Pythagorean Identity

sin²x + cos²x = 1.

Reciprocal Identities

sin x·csc x = 1, cos x·sec x = 1, tan x·cot x = 1.

Quotient Identities

tan x = sin x ⁄ cos x and cot x = cos x ⁄ sin x.

Even Function

cos (−x) = cos x; the graph is symmetric about the y-axis.

Odd Function

sin (−x) = −sin x and tan (−x) = −tan x; the graph is symmetric about the origin.

Cofunction Identity

sin (π⁄2 − x) = cos x, cos (π⁄2 − x) = sin x, etc.—functions of complementary angles.

Complementary Angles

Two angles whose sum is 90° (π⁄2 radians).

Double-Angle Formula for Sine

sin 2x = 2 sin x cos x.

Double-Angle Formula for Cosine

cos 2x = cos²x − sin²x = 2 cos²x − 1 = 1 − 2 sin²x.

Double-Angle Formula for Tangent

tan 2x = 2 tan x ⁄ (1 − tan²x).

Half-Angle Formula for Sine

sin (x⁄2) = ±√[(1 − cos x)⁄2].

Half-Angle Formula for Cosine

cos (x⁄2) = ±√[(1 + cos x)⁄2].

Half-Angle Formula for Tangent

tan (x⁄2) = sin x ⁄ (1 + cos x) = (1 − cos x) ⁄ sin x.

Triple-Angle Formula for Sine

sin 3x = 3 sin x − 4 sin³x.

Triple-Angle Formula for Tangent

tan 3x = (3 tan x − tan³x) ⁄ (1 − 3 tan²x).

Sum Formula for Sine

sin (A + B) = sin A cos B + cos A sin B.

Difference Formula for Sine

sin (A − B) = sin A cos B − cos A sin B.

Sum Formula for Cosine

cos (A + B) = cos A cos B − sin A sin B.

Difference Formula for Cosine

cos (A − B) = cos A cos B + sin A sin B.

Product-to-Sum (Factorization) for Sine

sin A + sin B = 2 sin[(A + B)/2] cos[(A − B)/2].

Product-to-Sum for Cosine

cos A + cos B = 2 cos[(A + B)/2] cos[(A − B)/2].

Sum-to-Product (Defactorization) for Sine

2 sin A cos B = sin (A + B) + sin (A − B).

Sum-to-Product for Cosine

2 cos A cos B = cos (A + B) + cos (A − B).

Principal Solution

The smallest positive solution (usually within 0 to 2π) of a trigonometric equation.

General Solution

All solutions to a trig equation, typically written using nπ or 2nπ to account for periodicity.

Period of Sine/Cosine

2π; the functions repeat every 360°.

Period of Tangent/Cotangent

π; the functions repeat every 180°.

Quadrant Sign Rule ("All Students Take Calculus")

In QI all trig functions are +; QII sin only +; QIII tan only +; QIV cos only +.

Reference Angle

The acute angle formed by the terminal side of an angle and the x-axis; used to find trig values in any quadrant.