| Date | None Specimen | Marks available | 2 | Reference code | SPNone.1.hl.TZ0.1 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Find | Question number | 1 | Adapted from | N/A |
Question
The angle \(\theta \) lies in the first quadrant and \(\cos \theta = \frac{1}{3}\).
Write down the value of \(\sin \theta \) .
[1]
a.
Find the value of \(\tan 2\theta \) .
[2]
b.
Find the value of \(\cos \left( {\frac{\theta }{2}} \right)\) , giving your answer in the form \(\frac{{\sqrt a }}{b}\) where a , \(b \in {\mathbb{Z}^ + }\) .
[3]
c.
Markscheme
\(\sin \theta = \frac{{\sqrt 8 }}{3}\) A1
[1 mark]
a.
\(\tan 2\theta = \frac{{2 \times \sqrt 8 }}{{1 - 8}} = - \frac{{2\sqrt 8 }}{7}\,\,\,\,\,\left( { - \frac{{4\sqrt 2 }}{7}} \right)\) M1A1
[2 marks]
b.
\({\cos ^2}\left( {\frac{\theta }{2}} \right) = \frac{{1 + \frac{1}{3}}}{2} = \frac{2}{3}\) M1A1
\(\cos \left( {\frac{\theta }{2}} \right) = \frac{{\sqrt 6 }}{3}\) A1
[3 marks]
c.
Examiners report
[N/A]
a.
[N/A]
b.
[N/A]
c.
