Pack B — Fluency

Mathematics

Year 11 · 11.8 Unit Circle

Pack B · Fluency

Name: _________________________________

Date: _________________ Class: ___________

Answer all questions. Show your working. Questions are grouped by challenge level.

BronzeQuestions 1–10

1.
Convert $135^\circ$ to radians, as an exact multiple of $\pi$ .
2.
Convert $\dfrac{ {a}\pi}{ {b}}$ radians to degrees.
3.
Find the exact value of $\sin\!\left(\dfrac{\pi}{ {n}}\right)$ .
4.
Find the exact value of $\cos\!\left(\dfrac{\pi}{ {n}}\right)$ .
5.
Find the exact value of $\tan\!\left(\dfrac{\pi}{ {n}}\right)$ .
6.
A sector has radius 8 cm and angle $1.5$ rad. Find the arc length.
7.
A sector has radius 10 cm and angle $0.6$ rad. Find its area.
8.
A sector has radius 8 cm and angle $45^\circ$ . Find the arc length, to 3 s.f.
9.
A sector has radius 10 cm and angle $60^\circ$ . Find the sector area, to 3 s.f.
10.
Using a 45-45-90 triangle, give the exact value of $\sin 45^\circ$ and $\cos 45^\circ$ .

SilverQuestions 11–20

11.
Find the exact value of $\cos\!\left(\dfrac{ {a}\pi}{ {b}}\right)$ .
12.
Find the exact value of $\sin\!\left(\dfrac{ {a}\pi}{ {b}}\right)$ .
13.
Solve $2\cos x = -1$ for $0 \leq x \leq 2\pi$ .
14.
Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in QI, find $\cos\theta$ exactly.
15.
If $\sin\theta = \dfrac{3}{5}$ and $\cos\theta = \dfrac{4}{5}$ , find $\tan\theta$ .
16.
Find the perimeter of a sector with radius 5 cm and angle $\dfrac{ {a}\pi}{ {b}}$ rad, to 3 s.f.
17.
State the value of $\sin\!\left(\dfrac{\pi}{2} - \dfrac{\pi}{6}\right)$ exactly.
18.
Simplify $\sin(-\theta)$ and $\cos(-\theta)$ .
19.
Solve $\tan x = -\sqrt{3}$ for $0 \leq x \leq 2\pi$ .
20.
A sector has radius 9 cm and angle $\dfrac{4\pi}{9}$ rad. Find (a) the arc length, (b) the sector area, exactly.

GoldQuestions 21–30

21.
Find an angle between $0$ and $2\pi$ coterminal with $\dfrac{13\pi}{4}$ .
22.
Find the exact value of $\sin\!\left(\dfrac{11\pi}{6}\right) + \cos\!\left(\dfrac{5\pi}{6}\right)$ .
23.
Solve $2\sin^2 x - 1 = 0$ for $0 \leq x \leq 2\pi$ .
24.
A chord subtends $\dfrac{2\pi}{3}$ rad at the centre of a circle of radius 10 cm. Find the area of the minor segment, to 3 s.f.
25.
Verify the identity $\dfrac{1 - \cos^2\theta}{\sin\theta} = \sin\theta$ .
26.
If $\sin\theta = -\dfrac{4}{5}$ and $\theta$ is in QIII, find $\cos\theta$ and $\tan\theta$ .
27.
Solve $\sin(2x) = \dfrac{1}{2}$ for $0 \leq x < 2\pi$ .
28.
Use $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ and a half-angle / difference identity to find $\sin\dfrac{\pi}{12}$ exactly.
29.
Convert $\dfrac{7\pi}{12}$ rad to degrees.
30.
Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x < 2\pi$ .

PlatinumQuestions 31–40

31.
A chord of length 12 cm subtends an angle of $\dfrac{\pi}{3}$ rad at the centre of a circle. Find the radius and the area of the minor segment, to 3 s.f.
32.
A circle has radius 8 cm and a chord whose arc has length 12 cm. Find the angle subtended at the centre, exact and to 3 s.f.
33.
Prove the identity $\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = \dfrac{2}{\cos\theta}$ .
34.
Solve $\cos 2x = \sin x$ for $0 \leq x \leq 2\pi$ .
35.
A sector has perimeter 20 cm. Find the radius that maximises the area.
36.
Find the exact value of $\sin\!\left(\dfrac{7\pi}{12}\right)$ using $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$ .
37.
A circle of radius 6 cm has a chord of length 6 cm. Find the angle subtended at the centre and the area of the minor segment, exact and to 3 s.f.
38.
Solve $\sin x + \cos x = 1$ for $0 \leq x \leq 2\pi$ .
39.
An annular sector has inner radius 4 cm, outer radius 7 cm, and angle $\dfrac{\pi}{3}$ rad. Find its area exactly.
40.
Show that if $\sin\theta + \cos\theta = \dfrac{1}{2}$ , then $\sin\theta \cos\theta = -\dfrac{3}{8}$ .