Hint

Choose R positive. Square and add the coefficient equations.

Worked solution

(a)

Comparing coefficients gives \(R\cos\alpha=12,R\sin\alpha=16\). Choose \(\displaystyle \alpha = \arctan (\frac{b}{a})\) in the first quadrant.

Answer: \(\displaystyle 20 \sin{\left(x + \operatorname{atan}{\left(\frac{4}{3} \right)} \right)}\)

(b)

The sine factor ranges from −1 to 1.

Answer: \(-8\) and \(32\)

(c)

Divide by R after subtracting D, so \(\displaystyle \sin (x+ \alpha )= \frac{1}{2}\). Use \(\displaystyle x+ \alpha = \frac{\pi }{6}\) or \(\displaystyle \frac{5\pi }{6}\) modulo 2π, subtract α, and retain exactly the representatives in [0,2π).

Answer: \(\displaystyle \left\{- \operatorname{atan}{\left(\frac{4}{3} \right)} + \frac{5 \pi}{6}, - \operatorname{atan}{\left(\frac{4}{3} \right)} + \frac{13 \pi}{6}\right\}\)

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: Compare sine and cosine coefficients and choose R>0 and the correct quadrant for α.
• 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: The coefficient of sinx is Rcosα.

(b) · 2 marks

• 1 mark: Use \(\displaystyle -1\le \sin (x+\alpha )\le 1\) and include the vertical translation.
• 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: A principal inverse-sine angle does not give every solution.

(c) · 2 marks

• 1 mark: Reduce to \(\displaystyle \sin (x+ \alpha )= \frac{1}{2}\), take both angle families and restrict x to the requested 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: The coefficient of sinx is Rcosα.

Total: 6 marks.