Q4. Write sin ωt + √3 cos ωt in the form D sin(ωt + φ).
A = 1 and B = √3, so D = √(1 + 3) = 2 and tan φ = √3, giving φ = π/3. The function is 2 sin(ωt + π/3).
Q5. Find the period of sin 2πt + sin 3πt, with t in seconds.
The terms have periods 2π/2π = 1 s and 2π/3π = 2/3 s. The smallest time that is a whole number of both is 2 s (2 × 1 s = 3 × 2/3 s), so the period is 2 s.
Q6. Mercury goes round the Sun once in 88 days. Find the frequency of this periodic motion in hertz.
T = 88 × 86 400 s, about 7.6 × 10 s, so ν = 1/T, about 1.3 × 10 Hz.