| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Given sin/cos/tan, find other expressions |
| Difficulty | Moderate -0.3 Part (i) is pure recall of a standard formula. Part (ii) is a routine application requiring finding cos α from sin α using Pythagorean identity, then substituting into the double angle formula. Part (iii) requires rewriting sec β and sin 2β in terms of sin β and cos β, then solving a resulting quadratic, but follows standard techniques. Overall slightly easier than average due to straightforward methodology and no novel problem-solving required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(\sin 2\theta = 2\sin\theta\cos\theta\) | B1 1 | or equiv; any letter acceptable here (and in parts (ii) and (iii)) |
| (ii) Attempt to find exact value of \(\cos\alpha\) | M1 | using identity attempt or right-angled triangle |
| Obtain \(\frac{1}{4}\sqrt{15}\) | A1 | or exact equiv |
| Substitute to confirm \(\frac{1}{4}\sqrt{15}\) | A1 3 | AG |
| (iii) State or imply \(\sec\beta = \frac{1}{\cos\beta}\) | B1 | |
| Use identity to produce equation involving \(\sin\beta\) | M1 | |
| Obtain \(\sin\beta = 0.3\) and hence \(17.5\) | A1 3 | and no other values between 0 and 90; allow 17.4 or value rounding to 17.4 or 17.5 |
**(i)** State $\sin 2\theta = 2\sin\theta\cos\theta$ | B1 1 | or equiv; any letter acceptable here (and in parts (ii) and (iii))
**(ii)** Attempt to find exact value of $\cos\alpha$ | M1 | using identity attempt or right-angled triangle
Obtain $\frac{1}{4}\sqrt{15}$ | A1 | or exact equiv
Substitute to confirm $\frac{1}{4}\sqrt{15}$ | A1 3 | AG
**(iii)** State or imply $\sec\beta = \frac{1}{\cos\beta}$ | B1 |
Use identity to produce equation involving $\sin\beta$ | M1 |
Obtain $\sin\beta = 0.3$ and hence $17.5$ | A1 3 | and no other values between 0 and 90; allow 17.4 or value rounding to 17.4 or 17.55 (i) Write down the identity expressing $\sin 2 \theta$ in terms of $\sin \theta$ and $\cos \theta$.\\
(ii) Given that $\sin \alpha = \frac { 1 } { 4 }$ and $\alpha$ is acute, show that $\sin 2 \alpha = \frac { 1 } { 8 } \sqrt { 15 }$.\\
(iii) Solve, for $0 ^ { \circ } < \beta < 90 ^ { \circ }$, the equation $5 \sin 2 \beta \sec \beta = 3$.
\hfill \mbox{\textit{OCR C3 2006 Q5 [7]}}