4400 Higher June 2009 Paper 4 Q20
20 The diagram shows an equilateral triangle of side 2 m.
Diagram NOT accurately drawn
(a)
(i) Use the diagram to show that \(\cos 60^\circ = \dfrac{1}{2}\)
(ii) Use the diagram to find the exact value of \(\sin 60^\circ\)
Give your answer as a surd. (4)
Give your answer as a surd. (4)
(b) Use the exact values of \(\cos 60^\circ\) and \(\sin 60^\circ\) to show that \((\cos 60^\circ)^2 + (\sin 60^\circ)^2 = 1\) (2)
| Scheme | Marks |
|---|---|
| (i) \(BM = 1\) or \(CM = 1\) | B1 |
| (ii) \((AM^2 =)\ 2^2 - 1^2\) (= 3) | M1 |
| \((AM =)\ \sqrt{2^2 - 1^2}\) (= \(\sqrt{3}\)) | M1 |
| \(\frac{\sqrt{3}}{2}\) or \(\sqrt{\frac{3}{4}}\) | A1 |
| (4) |
Notes
B1: (can be marked on diagram) allow cosine rule method
A1: (dependent on 1 line of Pythagoras or sine rule)
| Scheme | Marks |
|---|---|
| \(\left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2\) | M1 |
| \(= \frac{3}{4} + \frac{1}{4}\) oe | A1 |
| (2) | |
| (6 marks) |
Notes
M1: \(\left(\frac{\sqrt{3}}{2}\right)^2\) Must be seen
A1: allow 0.75 + 0.25 if M1 gained