APPC Unit 3C: Trigonometric Identities

Reid AP Precalculus

Reciprocal: sin x

sin x = 1/csc x.

The reciprocal identity states that the sine function is the reciprocal of the cosecant function.

Reciprocal: cos x

cos x = 1/sec x

The reciprocal identity states that the cosine function is the reciprocal of the secant function.

Reciprocal: tan x

tan x = 1/cot x

The reciprocal identity states that the tangent function is the reciprocal of the cotangent function.

Reciprocal: csc x

csc x = 1/sin x

The reciprocal identity states that the cosecant function is the reciprocal of the sine function.

Reciprocal: sec x

sec x = 1/cos x

The reciprocal identity states that the secant function is the reciprocal of the cosine function.

Reciprocal: cot x

cot x = 1/tan x

The reciprocal identity states that the cotangent function is the reciprocal of the tangent function.

Quotient: tan x

tan x = sin x / cos x

Quotient: cot x

cot x = cos x / sin x

Pythagorean: sin²x

sin²x = 1-cos²x

Pythagorean: cos²x

cos²x = 1-sin²x

Pythagorean: sin²x + cos²x

sin²x + cos²x = 1

Pythagorean: 1+tan²x

1+tan²x = sec²x

Pythagorean: 1+cot²

1+cot² = csc²x

Sum: sin(a+b)

sin(a+b) = sin a ⋅ cos b + cos a ⋅ sin b

Sum: cos(a+b)

cos(a+b) = cos a ⋅ cos b - sin a⋅ sin b

Difference: sin(a-b)

sin(a-b) = sin a ⋅ cos b - cos a ⋅ sin b

Difference: cos (a-b)

cos(a-b) = cos a ⋅ cos b + sin a ⋅ sin b

Double-Angle: sin(2u)

sin(2u) = 2sin u ⋅ cos u

Double-Angle: cos(2u)

cos(2u) =

⋅ cos²u - sin²u

⋅ 2cos²u-1

⋅ 1-2sin²u