Hint
The substituted sine value must be between −1 and 1. Factor in \(\displaystyle u=\sin (nx)\) and use the first valid angle.
Worked solution
(a)
Both u-values lie in the range of sine.
Answer: \((2u+1)(u-1)=0\); \(\displaystyle \left\{- \frac{1}{2}, 1\right\}\)
(b)
For each period, use \(\displaystyle nx= \frac{\pi }{2}, \frac{7\pi }{6}\) or \(\displaystyle \frac{11\pi }{6}\), then divide all angles by n.
Answer: \(\displaystyle \left\{\frac{\pi}{8}, \frac{7 \pi}{24}, \frac{11 \pi}{24}, \frac{5 \pi}{8}, \frac{19 \pi}{24}, \frac{23 \pi}{24}, \frac{9 \pi}{8}, \frac{31 \pi}{24}, \frac{35 \pi}{24}, \frac{13 \pi}{8}, \frac{43 \pi}{24}, \frac{47 \pi}{24}\right\}\)
(c)
There are three distinct angle solutions per full sine period.
\(12\)
Answer: \(12\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Factor the polynomial in u and retain the values in [−1,1].
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not keep a sine value outside its range.
(b) · 2 marks
- 1 mark: Solve every valid sine equation and list every root in the original interval.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not stop at the principal inverse-sine angle.
(c) · 2 marks
- 1 mark: There are three distinct angle solutions per full sine period.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not keep a sine value outside its range.
Total: 6 marks.