| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2002 |
| 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 straightforward trigonometric equation requiring standard identity manipulation (sin²x + cos²x = 1, tan x = sin x/cos x) followed by solving a quadratic in cos x. Part (i) is guided algebraic manipulation, and part (ii) is a routine application. Slightly easier than average due to the scaffolding provided in part (i). |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin x \tan x = (1 - \cos^2 x) + \cos x\) | B1 | Uses \(t = s/c\) and uses \(s^2 + c^2 = 1\) correctly |
| Answer | Marks | Guidance |
|---|---|---|
| \(\cos x = 0.5 \Rightarrow x = 60°\) or \(x = 300°\) | M1, DM1, A1, A1\* | Forms a 3 term quadratic in cosine; Solves = 0; Correct only; For \(360° - \) (his answer) – loses this if other answers in range 0 to 360. Needs M1 and DM1. Guesswork B2 B2 |
**(i)** $\sin x \tan x = \sin x \sin x + \cos x$
$\sin x \tan x = (1 - \cos^2 x) + \cos x$ | B1 | Uses $t = s/c$ and uses $s^2 + c^2 = 1$ correctly
**(ii)** $2\sin\tan x = 3 \Rightarrow 2c^2 + 3c - 2 = 0$
$\cos x = 0.5 \Rightarrow x = 60°$ or $x = 300°$ | M1, DM1, A1, A1\* | Forms a 3 term quadratic in cosine; Solves = 0; Correct only; For $360° - $ (his answer) – loses this if other answers in range 0 to 360. Needs M1 and DM1. Guesswork B2 B22 (i) Show that $\sin x \tan x$ may be written as $\frac { 1 - \cos ^ { 2 } x } { \cos x }$.\\
(ii) Hence solve the equation $2 \sin x \tan x = 3$, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2002 Q2 [5]}}