| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Standard trigonometric equations |
| Type | Deduce related solution |
| Difficulty | Standard +0.3 Part (a) requires using the Pythagorean identity to convert to a quadratic in sin x, then solving—a standard technique. Part (b) is a straightforward substitution using the solutions from (a). This is a typical multi-part trigonometric equation question requiring familiar methods but no novel insight, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
(a) Solve, for $-180° \leq x < 180°$, the equation
$$3 \sin^2 x + \sin x + 8 = 9 \cos^2 x$$
giving your answers to 2 decimal places. [4]
(b) Hence find the smallest positive solution of the equation
$$3\sin^2(2\theta - 30°) + \sin(2\theta - 30°) + 8 = 9 \cos^2(2\theta - 30°)$$
giving your answer to 2 decimal places. [2]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q12 [6]}}