Hint

Let \(\displaystyle u=\sin x. An\) extreme sine value has only one angle in a half-open full turn.

Worked solution

(a)

With \(\displaystyle u=\sin x, 2u^{2}+u-1=(2u-1)(u+1)\). Thus \(\displaystyle u= \frac{1}{2}\) or −1, both in the sine range.

Answer: \((2\sin x-1)(\sin x+1)=0\); \(\displaystyle \sin x= \frac{1}{2}\) or −1.

(b)

Sin \(\displaystyle x= \frac{1}{2}\) gives \(\displaystyle \frac{\pi }{6}\) and \(\displaystyle \frac{5\pi }{6}\). Sin \(\displaystyle x=-1\) gives only \(\displaystyle \frac{3\pi }{2}\) in this interval. There are three distinct roots.

Answer: \(\displaystyle x= \frac{\pi }{6}, \frac{5\pi }{6}, \frac{3\pi }{2}\).

(c)

The equation has period 2π, so two full turns give six roots. The endpoints have sine zero and make the original left side −1, so including 4π adds no root. The second-turn roots are \(\displaystyle \frac{13\pi }{6}, \frac{17\pi }{6}, \frac{7\pi }{2}\).

Answer: Six; neither 0 nor 4π is a root.

Maths4U indicative marking guidance

Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.

(a) · 2 marks

• 1 mark: Obtain the correct factorisation.
• 1 mark: State both admissible sine values.

Alternative methods: The quadratic formula is accepted.

Common error: Assuming a quadratic always gives four trigonometric angles.

(b) · 3 marks

• 1 mark: Give both roots for sine \(\displaystyle \frac{1}{2}\).
• 1 mark: Give the single root for sine −1.
• 1 mark: State the complete distinct solution set on the interval.

Alternative methods: A graph intersecting the two admissible sine levels is accepted.

Common error: Listing a second copy of the same minimum-angle root.

(c) · 2 marks

• 1 mark: Use the 2π period to obtain two sets of three roots.
• 1 mark: Give six and correctly check the included endpoints.

Alternative methods: A complete list of six valid roots is accepted.

Common error: Adding a seventh root solely because the upper endpoint is included.

Total: 7 marks.