| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Sine and Cosine Rules |
| Type | Exact trigonometric values |
| Difficulty | Moderate -0.8 This is a straightforward application of standard formulas with no problem-solving required. Part (i) is direct substitution into the compound angle formula with exact values from memory. Part (ii) applies the sine rule mechanically using the given result. Both parts are highly scaffolded with clear instructions on which formulas to use. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05l Double angle formulae: and compound angle formulae |
(i) Use the formula for $\sin(\theta + \phi)$, with $\theta = 45°$ and $\phi = 60°$, to show that $\sin 105° = \frac{\sqrt{3}+1}{2\sqrt{2}}$. [4]
(ii) In triangle ABC, angle BAC $= 45°$, angle ACB $= 30°$ and AB $= 1$ unit (see Fig. 3).
Using the sine rule, together with the result in part (i), show that $AC = \frac{\sqrt{3}+1}{2}$. [3]3 (i) Use the formula for $\sin ( \theta + \phi )$, with $\theta = 45 ^ { \circ }$ and $\phi = 60 ^ { \circ }$, to show that $\sin 105 ^ { \circ } = \frac { \sqrt { 3 } + 1 } { 2 \sqrt { 2 } }$.\\
(ii) In triangle ABC , angle $\mathrm { BAC } = 45 ^ { \circ }$, angle $\mathrm { ACB } = 30 ^ { \circ }$ and $\mathrm { AB } = 1$ unit (see Fig. 3).
Fig. 3
Using the sine rule, together with the result in part (i), show that $\mathrm { AC } = \frac { \sqrt { 3 } + 1 } { \sqrt { 2 } }$.
\hfill \mbox{\textit{OCR MEI C4 2007 Q3 [7]}}