| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2016 |
| Session | June |
| Marks | 5 |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Challenging +1.2 Part (i) requires systematic application of compound angle formulas and double angle identities with careful algebraic manipulation—more involved than routine A-level trig proofs. Part (ii) uses the proven identity to solve an equation, which is straightforward once the identity is established. The multi-step nature and algebraic complexity place this above average difficulty but within reach of strong A-level students. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
**(i)**
Use $\sin\left(\theta + \frac{\pi}{3}\right) = \sin\theta\cos\frac{\pi}{3} + \cos\theta\sin\frac{\pi}{3}$ **B1**
(Award even if in incorrect expansion of $\sin^2\left(\theta + \frac{\pi}{3}\right)$)
Expand $\sin^2\left(\theta + \frac{\pi}{3}\right)$ to obtain a term involving $\sin\theta\cos\theta$ **M1**
Use $\sin 2\theta = 2\sin\theta\cos\theta$ **B1**
Obtain $\frac{\sqrt{3}}{4}\sin 2\theta$ **AG** **A1**
**[4]**
*Alternative method:*
Use $\sin^2\theta = \frac{1}{2}(1 - \cos 2\theta)$ **B1**
Use $\cos\left(2\theta + \frac{2}{3}\pi\right) = \cos 2\theta\cos\frac{2}{3}\pi - \sin 2\theta\sin\frac{2}{3}\pi$ **B1**
Substitute and evaluate expression **M1**
Obtain $\frac{\sqrt{3}}{4}\sin 2\theta$ **AG** **A1**
**(ii)**
Use the result in (i) to obtain an equation in $\sin 2\theta$ **M1**
Obtain $\sin 2\theta = \frac{-1}{\sqrt{3}}$ **A1**
Use correct order of operations to obtain $\theta$ from an eqn in $\sin 2\theta$ **M1**
Obtain any two correct angles **A1**
Obtain answers rounding to $-0.308$, $2.83$, $-1.26$, $1.88$ **A1**
**[5]**11 (i) Prove that
$$\sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \frac { 1 } { 2 } \sin ^ { 2 } \theta - \frac { 3 } { 4 } = \frac { 1 } { 4 } \sqrt { 3 } \sin 2 \theta .$$
(ii) Hence solve the equation
$$2 \sin ^ { 2 } \left( \theta + \frac { 1 } { 3 } \pi \right) + \sin ^ { 2 } \theta = 1 \text { for } - \pi \leqslant \theta \leqslant \pi .$$
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2016 Q11 [5]}}