Module 5: Practice Problems
1
Verify: tanθ sinθ + cosθ = secθ
Solution
(sin²/cos) + cos = (sin²+cos²)/cos = 1/cos = secθ.
2
Find exact value of sin 75°.
Solution
sin(45+30) = (√6+√2)/4.
3
Find cos 2θ if sinθ = 4/5.
Solution
1 − 2sin² = 1 − 32/25 = −7/25.
4
Find sin 2θ if cosθ = −3/5 (θ in QII).
Solution
sinθ = 4/5. sin2θ = 2(4/5)(−3/5) = −24/25.
5
Write cos5x cos3x as a sum.
Solution
(1/2)(cos2x + cos8x).
6
Write sin7x + sin3x as a product.
Solution
2 sin5x cos2x.
7
Find tan(A+B) if tanA=2, tanB=3.
Solution
(2+3)/(1−6) = 5/(−5) = −1.
8
Verify: (sinA + cosA)² = 1 + sin2A.
Solution
sin²+2sincos+cos² = 1+2sincos = 1+sin2A.
9
Use half-angle to find sin22.5°.
Solution
√[(1−cos45)/2] = √[(1−√2/2)/2] = √[(2−√2)/4].
10
Simplify cos²x using a power-reducing formula.
Solution
(1 + cos2x)/2.