Compound angle identities; double angle for tan practice questions.

Using a compound angle identity, determine the exact value of cos 75°.

• (√6 + √2)/4
• (√6 − √2)/4
• (√2 − √6)/4
• (1 + √3)/4

Answer: (√6 − √2)/4

cos 75° = cos(45° + 30°) = cos 45° cos 30° − sin 45° sin 30° = (√6 − √2)/4. The plus version (√6 + √2)/4 is cos 15°, from using the wrong sign.

Identify the correct expansion of tan(A − B).

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

Answer: (tan A − tan B)/(1 + tan A tan B)

For tan(A − B) the numerator takes the minus sign and the denominator the plus sign: (tan A − tan B)/(1 + tan A tan B). Mixing the signs is the classic error.

Simplify sin 3x cos x − cos 3x sin x.

• sin 4x
• cos 2x
• cos 4x
• sin 2x

Answer: sin 2x

The expression matches sin(A − B) with A = 3x and B = x, so it equals sin(3x − x) = sin 2x. Reading it as sin(A + B) gives the distractor sin 4x.

Given that tan θ = 1/2, calculate the exact value of tan 2θ.

• 4/3
• 1
• 2/3
• 4/5

Answer: 4/3

tan 2θ = 2 tan θ/(1 − tan²θ) = 1/(1 − 1/4) = 1 ÷ (3/4) = 4/3. Answering 1 comes from doubling tan θ and forgetting the denominator.

By writing 75° as 45° + 30°, determine the exact value of tan 75°.

• 2 − √3
• 2 + √3
• 1 + √3
• (√3 + 1)/2

Answer: 2 + √3

tan 75° = (tan 45° + tan 30°)/(1 − tan 45° tan 30°) = (1 + 1/√3)/(1 − 1/√3) = (√3 + 1)/(√3 − 1) = 2 + √3 after rationalising. The value 2 − √3 is tan 15°.

A and B are acute angles with sin A = 3/5 and cos B = 5/13. Calculate the exact value of sin(A + B).

• 33/65
• 56/65
• 63/65
• 16/65

Answer: 63/65

With cos A = 4/5 and sin B = 12/13, sin(A + B) = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65. The value 56/65 is cos(A − B), a common mix-up.