Mathematical Methods — Question 14

QCAA 2025, Paper 1 · 4 marks

Q14 · 2025 · Technology-freeSimple familiar4 marks

QUESTION 14 (4 marks)

Determine all solutions to

2\cos(\theta)-1=0

, where

0^\circ\le\theta\le720^\circ

WORKED SOLUTION

4 marks

ANSWER

\theta=60^\circ,300^\circ,420^\circ,660^\circ

Worked solution

2\cos\theta-1=0\implies \cos\theta=\frac12.

The first-quadrant angle is

60^\circ

. Cosine is positive in quadrants 1 and 4, so in the first revolution

\theta=60^\circ,300^\circ.

For

0^\circ\le\theta\le720^\circ

\theta=60^\circ,300^\circ,420^\circ,660^\circ.

correctly rearranges the equation to make

\cos(\theta)

the subject

[1 mark]

determines the first quadrant (acute) angle in degrees

[1 mark]

determines the fourth quadrant angle in degrees

[1 mark]

determines the two remaining angles in the

360^\circ\le\theta\le720^\circ

domain

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.

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