Precalculus Trigonometry & Trigonometric Identities

Area of Triangle

A = ½ * a * b * sinθ

Law of Sines

SinA / a = SinB / b = SinC / c

Cosine Rule

C = a² + b² - 2 * a * b * cosC

Pythagorean Identity

sin²x + cos²x = 1

Cosine from Sine

cos²x = 1 - sin²x

Sine from Cosine

sin²x = 1 - cos²x

Tangent Identity

tan²x + 1 = sec²x

Cotangent Identity

1 + cot²x = csc²x

Odd Function for Sine

sin(-x) = -sin(x)

Even Function for Cosine

cos(-x) = cos(x)

Odd Function for Tangent

tan(-x) = -tan(x)

Cosecant Definition

cscx = 1/sinx

Secant Definition

secx = 1/cosx

Cotangent Definition

cotx = cosx/sinx

Tangent Definition

tanx = sinx/cosx

Cotangent in terms of Tangent

cotx = 1/tanx

Sine Sum Identity

sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b)

Cosine Sum Identity

cos(a ± b) = cos(a)cos(b) ∓ sin(a)sin(b)

Tangent Sum Identity

tan(a ± b) = (tan(a) ± tan(b)) / (1 ∓ tan(a)tan(b))

Double Angle for Sine

sin(2x) = 2sinxcosx

Double Angle for Cosine

cos(2x) = cos²x - sin²x

Double Angle for Cosine as Cosine

cos(2x) = 2cos²x - 1

Double Angle for Cosine as Sine

cos(2x) = 1 - 2sin²x

Double Angle for Tangent

tan(2x) = (2tanx) / (1 - tan²x)

Half-Angle for Sine

sin(x/2) = ± √(1 - cosx)/2

Half-Angle for Cosine

cos(x/2) = ± √(1 + cosx)/2

Half-Angle for Tangent

tan(x/2) = ± (1 - cosx)/sinx