C3 January 2007 Q1
1.
(a) By writing \(\sin 3\theta\) as \(\sin(2\theta + \theta)\), show that\[\sin 3\theta = 3\sin\theta - 4\sin^3\theta.\] (5)
(b) Given that \(\sin\theta = \dfrac{\sqrt{3}}{4}\), find the exact value of \(\sin 3\theta\). (2)
| Scheme | Marks |
|---|---|
| \(\sin 3\theta = \sin(2\theta + \theta) = \sin 2\theta\cos\theta + \cos 2\theta\sin\theta\) | B1 |
| \(= 2\sin\theta\cos^2\theta + (1 - 2\sin^2\theta)\sin\theta\) | B1 B1 |
| \(= 2\sin\theta - 2\sin^3\theta + \sin\theta - 2\sin^3\theta\) | M1 |
| \(= 3\sin\theta - 4\sin^3\theta\) * cso | A1 |
| (5) |
| Scheme | Marks |
|---|---|
| \(\sin 3\theta = 3 \times \dfrac{\sqrt{3}}{4} - 4\left(\dfrac{\sqrt{3}}{4}\right)^3 = \dfrac{3\sqrt{3}}{4} - \dfrac{3\sqrt{3}}{16} = \dfrac{9\sqrt{3}}{16}\) or exact equivalent | M1 A1 |
| (2) | |
| (7 marks) |