3.7 Circular functions questions for IB Maths AA

61 questions on Circular functions, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 3 Geometry and trigonometry
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	16	3	19
Paper 2	33	9	42
Every paper	49	12	61

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00139Paper 1 · 6 marks

The depth of water, $d$ metres, at the entrance to a harbour is modelled by the function

d (t) = 7 + 4 \sin (\frac{π t}{6}), 0 \leq t \leq 12,

where $t$ is the time in hours after midnight.

(a)
Find the depth of water at 07:00.
2 marks
(b)
Find the exact depth of water at 10:00.
2 marks
(c)
Find the maximum depth of water and the time at which it occurs.
2 marks

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Question 2 · AA-00160Paper 2 · 7 marks

Let $f (x) = 6 \sin x \cos x$ , for $0 \leq x \leq π$ .

(a)
Express $f (x)$ in the form $a \sin (b x)$ , where $a, b \in Z^{+}$ .
2 marks
(b)(i)
Write down the period of $f$ .
1 mark
(b)(ii)
Write down the range of $f$ .
1 mark
(c)
Solve the equation $f (x) = 1.8$ .
3 marks

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Question 3 · AA-00146Paper 1 · 16 marks

A seat on an observation wheel starts at its lowest point. The height of the seat above the ground, $h$ metres, $t$ minutes later is modelled by

h (t) = 20 - 18 \cos (\frac{π t}{12}), 0 \leq t \leq 24.

(a)
Find the exact height of the seat when $t = 9$ .
2 marks
(b)
Write down the maximum height of the seat and the value of $t$ at which it occurs.
2 marks
(c)
Find the values of $t$ for which $h (t) = 29$ .
4 marks
(d)
Hence, find the length of time during one revolution for which the seat is more than 29 m above the ground.
2 marks
(e)
Find the rate at which the height of the seat is changing when $t = 2$ .
3 marks
(f)
Find the values of $t$ at which the seat is descending at a rate of $\frac{3 π}{4}$ metres per minute.
3 marks

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Question 4 · AA-00166Paper 2 · 16 marks

The height, $h$ metres, of a buoy above the seabed $t$ hours after midnight is modelled by

h (t) = 4 + 3 \sin^{2} (\frac{π t}{12}), 0 \leq t \leq 24.

(a)
Show that $h (t) = 5.5 - 1.5 \cos (\frac{π t}{6})$ .
3 marks
(b)(i)
Write down the period of $h$ .
1 mark
(b)(ii)
Write down the maximum height and the minimum height of the buoy above the seabed.
2 marks
(c)
Find the total time during the 24 hours for which the buoy is more than 6.2 m above the seabed.
4 marks
(d)
Find the maximum rate of increase of the height of the buoy, and the first time at which it occurs.
3 marks
(e)
Find the values of $t$ for which the height of the buoy is increasing at a rate of 0.5 m per hour.
3 marks

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Question 5 · AA-00185Paper 1 · HL only · 18 marks

Consider the function $f (x) = 2 \cos (x - \frac{π}{6}) \sin (x + \frac{π}{3})$ , for $0 \leq x \leq π$ .

(a)
Show that $\cos (x - \frac{π}{6}) \equiv \sin (x + \frac{π}{3})$ .
3 marks
(b)
Hence show that $f (x) = 1 - \cos (2 x + \frac{2 π}{3})$ .
3 marks
(c)
Write down the range of $f$ .
1 mark
(d)
Describe a sequence of transformations that maps the graph of $y = \cos x$ onto the graph of $y = f (x)$ .
4 marks
(e)
Solve the equation $f (x) = \frac{1}{2}$ .
3 marks
(f)
Find the exact area between the graph of $f$ and the $x$ -axis, from $x = 0$ to $x = \frac{π}{2}$ .
4 marks

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Question 6 · AA-00177Paper 2 · HL only · 16 marks

A Ferris wheel has radius 20 m and its centre is 22 m above the ground. The wheel rotates at a constant speed and completes one revolution every 12 minutes.

Two cabins, P and Q, are attached to the rim of the wheel. At time $t = 0$ minutes, cabin P is at its lowest point. Cabin Q is ahead of cabin P, so that the angle at the centre of the wheel between the two cabins is $\frac{π}{3}$ .

The heights above the ground, in metres, of cabins P and Q at time $t$ minutes are given by

h_{P} (t) = 22 - 20 \cos (\frac{π t}{6}) and h_{Q} (t) = 22 - 20 \cos (\frac{π t}{6} + \frac{π}{3}), for t \geq 0.

(a)
Write down the maximum height of cabin P and the first time at which it reaches this height.
2 marks
(b)
Let $d (t) = h_{Q} (t) - h_{P} (t)$ . Show that $d (t) = 10 \cos (\frac{π t}{6}) + 10 \sqrt{3} \sin (\frac{π t}{6})$ .
3 marks
(c)
By expanding $20 \sin (\frac{π t}{6} + \frac{π}{6})$ , show that $d (t) = 20 \sin (\frac{π t}{6} + \frac{π}{6})$ .
3 marks
(d)
Write down the greatest value of $d (t)$ , and find the first time at which it occurs.
3 marks
(e)
Find the length of time, during the first revolution, for which cabin Q is more than 15 m higher than cabin P.
3 marks
(f)
Find the rate of change of $d$ when $t = 5$ .
2 marks

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