| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Moderate -0.3 This is a straightforward application of compound angle formulae to simplify a trigonometric equation. Part (i) requires expanding sin(x-60°) and cos(30°-x) using standard formulae, then algebraic manipulation to reach cos x = k. Part (ii) is routine solving within a given range. The question is slightly easier than average as it's a guided, standard textbook exercise with no novel insight required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use trig formulae to express LHS in terms of sin x and cos x | M1 | Use cos 60° = sin 30° to reduce equation to given form cos x = k |
| (ii) State or imply that \(k = -\frac{1}{\sqrt{3}}\) (accept -0.577 or -0.58) | A1 | Obtain answer \(x = 125.3°\) only. Answer must be in degrees; ignore answers outside the given range. |
**(i)** Use trig formulae to express LHS in terms of sin x and cos x | M1 | Use cos 60° = sin 30° to reduce equation to given form cos x = k | M1 | **[2]**
**(ii)** State or imply that $k = -\frac{1}{\sqrt{3}}$ (accept -0.577 or -0.58) | A1 | Obtain answer $x = 125.3°$ only. Answer must be in degrees; ignore answers outside the given range. | A1 | [SR: if $k = \frac{1}{\sqrt{3}}$ is followed by $x = 54.7°$, give A0A1√.] | **[2]**1 (i) Show that the equation
$$\sin \left( x - 60 ^ { \circ } \right) - \cos \left( 30 ^ { \circ } - x \right) = 1$$
can be written in the form $\cos x = k$, where $k$ is a constant.\\
(ii) Hence solve the equation, for $0 ^ { \circ } < x < 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2003 Q1 [4]}}