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Date None Specimen Marks available 1 Reference code SPNone.1.hl.TZ0.1
Level HL only Paper 1 Time zone TZ0
Command term Write down 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.

Syllabus sections

Topic 3 - Core: Circular functions and trigonometry » 3.2 » Definition of \(\cos \theta \) , \(\sin \theta \) and \(\tan \theta \) in terms of the unit circle.

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