| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation only (no integral) |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring proof of a non-trivial trigonometric identity using compound angle formulae, then applying it in three different contexts including finding exact values, solving equations, and determining range restrictions. While the techniques are all C3 standard, the identity proof requires careful algebraic manipulation through multiple steps, and part (iv) requires understanding of the range of trigonometric functions. This is more demanding than a typical C3 question but doesn't require exceptional insight. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State at least one of \(\cos\theta \cos 60 - \sin\theta \sin 60\) and \(\cos\theta \cos 30 - \sin\theta \sin 30\) | B1 | |
| Attempt complete multiplication of identities of form \(\pm\cos\cos \pm \sin\sin\) | M1 | with values \(\frac{1}{2}\sqrt{3}, \frac{1}{4}\) involved |
| Use \(\cos^2\theta + \sin^2\theta = 1\) and \(2\sin\theta\cos\theta = \sin 2\theta\) | M1 | |
| Obtain \(\sqrt{3} - 2\sin 2\theta\) | A1 4 | AG; necessary detail required |
| (ii) Attempt use of 22.5 in right-hand side | M1 | |
| Obtain \(\sqrt{3} - \sqrt{2}\) | A1 2 | or exact equiv |
| (iii) Obtain 10.7 | B1 | or greater accuracy; allow \(\pm0.1\) |
| Attempt correct process to find two angles | M1 | from values of \(2\theta\) between 0 and 180 |
| Obtain 79.3 | A1 3 | or greater accuracy and no others between 0 and 90; allow \(\pm0.1\) |
| (iv) Indicate or imply that critical values of \(\sin 2\theta\) are \(-1\) and \(1\) | M1 | |
| Obtain both of \(k > \sqrt{3} + 2, k < \sqrt{3} - 2\) | A1 | condoning decimal equivs, \(\leq\) signs |
| Obtain complete correct solution | A1 3 | now with exact values and unambiguously stated |
**(i)** State at least one of $\cos\theta \cos 60 - \sin\theta \sin 60$ and $\cos\theta \cos 30 - \sin\theta \sin 30$ | B1 |
Attempt complete multiplication of identities of form $\pm\cos\cos \pm \sin\sin$ | M1 | with values $\frac{1}{2}\sqrt{3}, \frac{1}{4}$ involved
Use $\cos^2\theta + \sin^2\theta = 1$ and $2\sin\theta\cos\theta = \sin 2\theta$ | M1 |
Obtain $\sqrt{3} - 2\sin 2\theta$ | A1 4 | AG; necessary detail required
**(ii)** Attempt use of 22.5 in right-hand side | M1 |
Obtain $\sqrt{3} - \sqrt{2}$ | A1 2 | or exact equiv
**(iii)** Obtain 10.7 | B1 | or greater accuracy; allow $\pm0.1$
Attempt correct process to find two angles | M1 | from values of $2\theta$ between 0 and 180
Obtain 79.3 | A1 3 | or greater accuracy and no others between 0 and 90; allow $\pm0.1$
**(iv)** Indicate or imply that critical values of $\sin 2\theta$ are $-1$ and $1$ | M1 |
Obtain both of $k > \sqrt{3} + 2, k < \sqrt{3} - 2$ | A1 | condoning decimal equivs, $\leq$ signs
Obtain complete correct solution | A1 3 | now with exact values and unambiguously stated9 (i) Use the identity for $\cos ( A + B )$ to prove that
$$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) \equiv \sqrt { 3 } - 2 \sin 2 \theta .$$
(ii) Hence find the exact value of $4 \cos 82.5 ^ { \circ } \cos 52.5 ^ { \circ }$.\\
(iii) Solve, for $0 ^ { \circ } < \theta < 90 ^ { \circ }$, the equation $4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = 1$.\\
(iv) Given that there are no values of $\theta$ which satisfy the equation
$$4 \cos \left( \theta + 60 ^ { \circ } \right) \cos \left( \theta + 30 ^ { \circ } \right) = k ,$$
determine the set of values of the constant $k$.
\hfill \mbox{\textit{OCR C3 2008 Q9 [12]}}