| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: tan/sin/cos identity manipulation |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation of a trigonometric equation requiring basic rearrangement to isolate tan θ, followed by routine solution within a given range. Part (i) is shown for the student, and part (ii) requires only calculator work and knowledge of the tangent function's periodicity—simpler than typical multi-step trigonometric problems. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(s + c = 2s - 2c \Rightarrow s = 3c \Rightarrow \tan\theta = 3\) | M1 A1 [2] | Use of \(t = s/c\) collection \(\Rightarrow \tan\theta = k\). Algebra needed to reduce to this form. |
| (ii) \(\Rightarrow \theta = 71.6°\) or \(251.6°\) | B1 B1 \(\checkmark\) [2] | B1 \(\checkmark\) for \(180° + \ldots\) as only soln in range. |
(i) $s + c = 2s - 2c \Rightarrow s = 3c \Rightarrow \tan\theta = 3$ | M1 A1 [2] | Use of $t = s/c$ collection $\Rightarrow \tan\theta = k$. Algebra needed to reduce to this form.
(ii) $\Rightarrow \theta = 71.6°$ or $251.6°$ | B1 B1 $\checkmark$ [2] | B1 $\checkmark$ for $180° + \ldots$ as only soln in range.3 (i) Show that the equation $\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )$ can be expressed as $\tan \theta = 3$.\\
(ii) Hence solve the equation $\sin \theta + \cos \theta = 2 ( \sin \theta - \cos \theta )$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2005 Q3 [4]}}