Trigonometric Identities and Relationships

Vocabulary flashcards covering key trigonometric identities and relationships extracted from the lecture notes.

Sine in terms of tangent

sin θ = tan θ / √(1 + tan² θ)

Cosine in terms of tangent

cos θ = 1 / √(1 + tan² θ)

Tangent in terms of sine

tan θ = sin θ / √(1 − sin² θ)

Cotangent in terms of cosine

cot θ = √(1 − cos² θ) / cos θ = sin θ / cos θ

Co-function identity for sine

sin(π/2 − θ) = cos θ

Co-function identity for cosine

cos(π/2 − θ) = sin θ

Co-function identity for tangent

tan(π/2 − θ) = cot θ

Supplementary angle identity for sine

sin(π − θ) = sin θ

Supplementary angle identity for cosine

cos(π − θ) = −cos θ

Supplementary angle identity for tangent

tan(π − θ) = −tan θ

Double-angle formula for sine

sin 2θ = 2 sin θ cos θ

Double-angle formula for cosine

cos 2θ = cos² θ − sin² θ = 1 − 2 sin² θ = 2 cos² θ − 1

Double-angle formula for tangent

tan 2θ = (2 tan θ) / (1 − tan² θ)

Triple-angle formula for sine

sin 3θ = 3 sin θ − 4 sin³ θ

Triple-angle formula for cosine

cos 3θ = 4 cos³ θ − 3 cos θ

Triple-angle formula for tangent

tan 3θ = (3 tan θ − tan³ θ) / (1 − 3 tan² θ)

Difference of sines to sum of cosines

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

Product-to-sum for cosine product

cos A · cos B = ½ [cos(A + B) + cos(A − B)]

Product-to-sum for sine–cosine

sin A · cos B = ½ [sin(A + B) + sin(A − B)]

Sum-to-product for sum of sines

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

Sum-to-product for difference of sines

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

Sum-to-product for sum of cosines

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

Sum-to-product for difference of cosines

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

Sine addition formula

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

Sine subtraction formula

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

Cosine addition formula

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

Cosine subtraction formula

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

Tangent addition formula

tan(A + B) = (tan A + tan B) / (1 − tan A tan B)

Tangent subtraction formula

tan(A − B) = (tan A − tan B) / (1 + tan A tan B)