| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: tan/sin/cos identity manipulation |
| Difficulty | Moderate -0.3 This is a standard two-part trigonometric equation question requiring conversion using tan²θ = sin²θ/cos²θ and the Pythagorean identity, followed by solving a quadratic in cos θ. The algebraic manipulation is straightforward and the solution method is routine for P1 level, making it slightly easier than average but not trivial due to the multi-step process and need to find all solutions in the given range. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Replaces \(\tan^2 \theta\) by \(\frac{\sin^2 \theta}{\cos^2 \theta}\) and | M1 | For correct formula used |
| Replaces \(\sin^2 \theta\) by \(1 - \cos^2 \theta\) | M1 | For correct formula used |
| \(\rightarrow 2\cos^2 \theta + 3\cos \theta - 2 = 0\) | [2] | |
| (ii) Soln of quadratic \(\rightarrow \frac{1}{2}\) and \(-2\) | M1 | Correct method of solving quadratic |
| \(\rightarrow 60°\) and \(300°\) | A1 A1√ | A1 for 60, A1√ for (360 – 1st answer) and no other solutions in the range. [3] |
**(i)** $2\tan^2 \theta \cos \theta = 3$
Replaces $\tan^2 \theta$ by $\frac{\sin^2 \theta}{\cos^2 \theta}$ and | M1 | For correct formula used
Replaces $\sin^2 \theta$ by $1 - \cos^2 \theta$ | M1 | For correct formula used
$\rightarrow 2\cos^2 \theta + 3\cos \theta - 2 = 0$ | [2]
**(ii)** Soln of quadratic $\rightarrow \frac{1}{2}$ and $-2$ | M1 | Correct method of solving quadratic
$\rightarrow 60°$ and $300°$ | A1 A1√ | A1 for 60, A1√ for (360 – 1st answer) and no other solutions in the range. [3]2 (i) Show that the equation $2 \tan ^ { 2 } \theta \cos \theta = 3$ can be written in the form $2 \cos ^ { 2 } \theta + 3 \cos \theta - 2 = 0$.\\
(ii) Hence solve the equation $2 \tan ^ { 2 } \theta \cos \theta = 3$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2008 Q2 [5]}}