Date | May 2014 | Marks available | 2 | Reference code | 14M.1.hl.TZ1.5 |
Level | HL only | Paper | 1 | Time zone | TZ1 |
Command term | Prove that | Question number | 5 | Adapted from | N/A |
Question
Use the identity cos2θ=2cos2θ−1 to prove that cos12x=√1+cosx2, 0⩽.
Find a similar expression for \sin \frac{1}{2}x,{\text{ }}0 \leqslant x \leqslant \pi .
Hence find the value of \int_0^{\frac{\pi }{2}} {\left( {\sqrt {1 + \cos x} + \sqrt {1 - \cos x} } \right){\text{d}}x} .
Markscheme
\cos x = 2{\cos ^2}\frac{1}{2}x - 1
\cos \frac{1}{2}x = \pm \sqrt {\frac{{1 + \cos x}}{2}} M1
positive as 0 \leqslant x \leqslant \pi R1
\cos \frac{1}{2}x = \sqrt {\frac{{1 + \cos x}}{2}} AG
[2 marks]
\cos 2\theta = 1 - 2{\sin ^2}\theta (M1)
\sin \frac{1}{2}x = \sqrt {\frac{{1 - \cos x}}{2}} A1
[2 marks]
\sqrt 2 \int_0^{\frac{\pi }{2}} {\cos \frac{1}{2}x + \sin \frac{1}{2}x{\text{d}}x} A1
= \sqrt 2 \left[ {2\sin \frac{1}{2}x - 2\cos \frac{1}{2}x} \right]_0^{\frac{\pi }{2}} A1
= \sqrt 2 (0) - \sqrt 2 (0 - 2) A1
= 2\sqrt 2 A1
[4 marks]