Trigonometric Identities

If A + B + C = π, state the identity for the sum of cosines: cos A + cos B + cos C

1 + 4 sin(A/2) sin(B/2) sin(C/2)

If A + B + C = π, what is the identity for cos(2A) + cos(2B) + cos(2C)?

-1 - 4 cos A cos B cos C

If A + B + C = π, what is the identity for sin(2A) + sin(2B) + sin(2C)?

4 sin A sin B sin C

What is the exact value of cos 15°?

(√3 + 1) / (2√2)

If A + B + C = π, state the identity for the sum of sines: sin A + sin B + sin C

4 cos(A/2) cos(B/2) cos(C/2)

What is the compound angle identity for sin(A + B)?

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

What is the compound angle identity for sin(A − B)?

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

What is the compound angle identity for cos(A + B)?

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

What is the compound angle identity for cos(A − B)?

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

What is the compound angle identity for tan(A + B)?

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

What is the compound angle identity for tan(A − B)?

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

What is the compound angle identity for cot(A + B)?

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

What is the compound angle identity for cot(A − B)?

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

Formula: 2 sin A cos B

sin(A + B) + sin(A − B)

Formula: 2 cos A cos B

cos(A + B) + cos(A − B)

Formula: 2 cos A sin B

sin(A + B) − sin(A − B)

Formula: 2 sin A sin B

cos(A − B) − cos(A + B)

Express sin²A − sin²B as a product of two sine functions

sin(A + B) sin(A − B)

Express cos²A − sin²B as a product of two cosine functions

cos(A + B) cos(A − B)

What is the sum-to-product identity for sin C + sin D?

sin C + sin D = 2 sin((C + D)/2) cos((C − D)/2)

What is the sum-to-product identity for sin C − sin D?

sin C − sin D = 2 cos((C + D)/2) sin((C − D)/2)

What is the sum-to-product identity for cos C + cos D?

cos C + cos D = 2 cos((C + D)/2) cos((C − D)/2)

What is the sum-to-product identity for cos C − cos D?

cos C − cos D = −2 sin((C + D)/2) sin((C − D)/2)

State the double angle identity for sin(2A) in terms of sin A and cos A

sin(2A) = 2 sin A cos A

State the double angle identity for sin(2A) in terms of tan A

sin(2A) = 2 tan A / (1 + tan²A)

What is the primary double angle identity for cos(2A) using both sine and cosine?

cos(2A) = cos²A − sin²A

State the double angle identity for cos(2A) using only sine

cos(2A) = 1 − 2 sin²A

State the double angle identity for cos(2A) using only cosine

cos(2A) = 2 cos²A − 1

State the double angle identity for cos(2A) in terms of tan A

cos(2A) = (1 − tan²A) / (1 + tan²A)

What is the double angle identity for tan(2A)?

tan(2A) = 2 tan A / (1 − tan²A)

Express cos²A in terms of cos(2A)

cos²A = (1 + cos(2A)) / 2

Express sin²A in terms of cos(2A)

sin²A = (1 − cos(2A)) / 2

Simplify (sin θ + cos θ)² using a double angle identity

1 + sin(2θ)

Simplify (sin θ − cos θ)² using a double angle identity

1 − sin(2θ)

What is the result of (1 + cos(2θ)) / sin(2θ)?

cot θ

What is the result of (1 − cos(2θ)) / sin(2θ)?

tan θ

Express sin θ in terms of tan(θ/2)

sin θ = 2 tan(θ/2) / (1 + tan²(θ/2))

Express cos θ in terms of tan(θ/2)

cos θ = (1 − tan²(θ/2)) / (1 + tan²(θ/2))

State the identity for tan θ in terms of tan(θ/2)

tan θ = 2 tan(θ/2) / (1 − tan²(θ/2))

Formula: tan(θ/2) in terms of sin θ and cos θ

tan(θ/2) = (1 − cos θ) / sin θ

Formula: cot(θ/2) in terms of sin θ and cos θ

cot(θ/2) = (1 + cos θ) / sin θ

What is the exact value of sin 18°?

(√5 − 1) / 4

What is the exact value of cos 36°?

(√5 + 1) / 4

What is the exact value of sin 15°?

(√3 − 1) / (2√2)

What is the exact value of tan 15°?

2 − √3

What is the exact value of tan 75°?

2 + √3

What is the exact value of sin 36°?

√(10 − 2√5) / 4

What is the exact value of sin 72°?

√(10 + 2√5) / 4

What is the range of the function f(θ) = a sin θ + b cos θ?

[−√(a² + b²), √(a² + b²)]

In the reduction of a sin θ + b cos θ, how is the auxiliary angle φ defined?

tan φ = b / a

What is the triple angle identity for sin(3A)?

sin(3A) = 3 sin A − 4 sin³A

What is the triple angle identity for cos(3A)?

cos(3A) = 4 cos³A − 3 cos A

What is the triple angle identity for tan(3A)?

tan(3A) = (3 tan A − tan³A) / (1 − 3 tan²A)

Simplify: sin θ sin(60° − θ) sin(60° + θ)

(1/4) sin(3θ)

Simplify: cos θ cos(60° − θ) cos(60° + θ)

(1/4) cos(3θ)

Simplify: tan θ tan(60° − θ) tan(60° + θ)

tan(3θ)

If A + B + C = π, what is the conditional identity for tan A + tan B + tan C?

tan A + tan B + tan C = tan A tan B tan C

If A + B + C = π, what is the conditional identity for cot A cot B + cot B cot C + cot C cot A?

cot A cot B + cot B cot C + cot C cot A = 1

If A + B + C = π, what does tan(A/2)tan(B/2) + tan(B/2)tan(C/2) + tan(C/2)tan(A/2) equal?

1

If A + B + C = π, what is cot(A/2) + cot(B/2) + cot(C/2) equal to?

cot(A/2) cot(B/2) cot(C/2)

What is the value of (1 + tan A)(1 + tan B) given that A + B = π/4?

2

Express cot θ − tan θ as a single trigonometric term

2 cot(2θ)

In the generalised tangent sum formula, what do the terms S₁ and S₂ represent?

S₁ = Σ tan θᵢ and S₂ = Σ(tan θᵢ tan θⱼ), where i < j

Identity for sum of tangents at intervals of 60°

tan A + tan(A + 60°) + tan(A + 120°) = 3 tan 3

cos(θ₁ + θ₂ + θ₃ + ⋯ + θₙ) = ?

(cos θ₁ cos θ₂ cos θ₃ ⋯ cos θₙ)(1 − S₂ + S₄ − S₆ + ⋯)

sin(θ₁ + θ₂ + θ₃ + ⋯ + θₙ) = ?

(cos θ₁ cos θ₂ cos θ₃ ⋯ cos θₙ)(S₁ − S₃ + S₅ − S₇ + ⋯)

tan(θ₁ + θ₂ + θ₃ + ⋯ + θₙ) = ?

(S₁ − S₃ + S₅ − S₇ + ⋯)

/ (1 − S₂ + S₄ − S₆ + ⋯)