| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 Part (i) requires systematic application of double angle formulae (cos 4θ = 2cos²2θ - 1, cos 2θ = 2cos²θ - 1) with algebraic manipulation—standard technique for P3. Part (ii) is straightforward substitution leading to 8cos⁴θ = 5, then solving cos²θ = √(5/8) for θ in the given range. This is a routine multi-step question testing standard identities and equation-solving with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Either: Express \(\cos 4\theta\) in terms of \(\cos 2\theta\) and/or \(\sin 2\theta\) | B1 | |
| Use double angle formulae to express LHS in terms of \(\cos \theta\) (and maybe \(\sin \theta\)) | M1 | |
| Obtain any correct expression in terms of \(\cos \theta\) alone | A1 | |
| Reduce correctly to the given form | A1 | |
| Or: Use double angle formula to express RHS in terms of \(\cos 2\theta\) | M1 | |
| Express \(\cos^2 2\theta\) in terms of \(\cos 4\theta\) | B1 | |
| Obtain any correct expression in terms of \(\cos 4\theta\) and \(\cos 2\theta\) | A1 | |
| Reduce correctly to the given form | A1 | 4 |
| (ii) Using the identity, carry out method for calculating one root | M1 | |
| Obtain answer \(27.2°\) (or \(0.475\) radians) or \(27.3°\) (or \(0.476\) radians) | A1 | |
| Obtain a second answer, e.g. \(332.8°\) (or \(5.81\) radians) | A1√ | |
| Obtain remaining answers, e.g. \(152.8°\) and \(207.2°\) (or \(2.67\) and \(3.62\) radians) and no others in range | A1√ | 4 |
**(i)** **Either:** Express $\cos 4\theta$ in terms of $\cos 2\theta$ and/or $\sin 2\theta$ | B1 |
Use double angle formulae to express LHS in terms of $\cos \theta$ (and maybe $\sin \theta$) | M1 |
Obtain any correct expression in terms of $\cos \theta$ alone | A1 |
Reduce correctly to the given form | A1 |
**Or:** Use double angle formula to express RHS in terms of $\cos 2\theta$ | M1 |
Express $\cos^2 2\theta$ in terms of $\cos 4\theta$ | B1 |
Obtain any correct expression in terms of $\cos 4\theta$ and $\cos 2\theta$ | A1 |
Reduce correctly to the given form | A1 | 4
**(ii)** Using the identity, carry out method for calculating one root | M1 |
Obtain answer $27.2°$ (or $0.475$ radians) or $27.3°$ (or $0.476$ radians) | A1 |
Obtain a second answer, e.g. $332.8°$ (or $5.81$ radians) | A1√ |
Obtain remaining answers, e.g. $152.8°$ and $207.2°$ (or $2.67$ and $3.62$ radians) and no others in range | A1√ | 46 (i) Prove the identity
$$\cos 4 \theta + 4 \cos 2 \theta \equiv 8 \cos ^ { 4 } \theta - 3$$
(ii) Hence solve the equation
$$\cos 4 \theta + 4 \cos 2 \theta = 2$$
for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P3 2005 Q6 [8]}}