Answer Key
11.8 Unit Circle
Pack A — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Convert $ 60^\circ$ to radians, as an exact multiple of $\pi$. | $\dfrac{\pi}{3}$ |
| 2 | Convert $\dfrac{ 5\pi}{ 6}$ radians to degrees. | $150^\circ$ |
| 3 | Find the exact value of $\sin\!\left(\dfrac{\pi}{ 6}\right)$. | $\dfrac{1}{2}$ |
| 4 | Find the exact value of $\cos\!\left(\dfrac{\pi}{ 3}\right)$. | $\dfrac{1}{2}$ |
| 5 | Find the exact value of $\tan\!\left(\dfrac{\pi}{ 4}\right)$. | 1 |
| 6 | A sector has radius 6 cm and angle $ 2$ rad. Find the arc length. | 12 cm |
| 7 | A sector has radius 6 cm and angle $ 2$ rad. Find its area. | 36 cm$^2$ |
| 8 | A sector has radius 10 cm and angle $ 72^\circ$. Find the arc length, to 3 s.f. | 12.6 cm |
| 9 | A sector has radius 8 cm and angle $ 45^\circ$. Find the sector area, to 3 s.f. | 25.1 cm$^2$ |
| 10 | Using a 45-45-90 triangle, give the exact value of $\sin 45^\circ$ and $\cos 45^\circ$. | Both equal $\dfrac{\sqrt{2}}{2}$. |
| 11 | Find the exact value of $\cos\!\left(\dfrac{ 5\pi}{ 6}\right)$. | $-\dfrac{\sqrt{3}}{2}$ |
| 12 | Find the exact value of $\sin\!\left(\dfrac{ 7\pi}{ 6}\right)$. | $-\dfrac{1}{2}$ |
| 13 | Solve $2\cos x = -1$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{2\pi}{3}$ or $\dfrac{4\pi}{3}$ |
| 14 | Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in QI, find $\cos\theta$ exactly. | $\dfrac{4}{5}$ |
| 15 | If $\sin\theta = \dfrac{3}{5}$ and $\cos\theta = \dfrac{4}{5}$, find $\tan\theta$. | $\dfrac{3}{4}$ |
| 16 | Find the perimeter of a sector with radius 9 cm and angle $\dfrac{ 4\pi}{ 9}$ rad, to 3 s.f. | 30.6 cm |
| 17 | State the value of $\sin\!\left(\dfrac{\pi}{2} - \dfrac{\pi}{6}\right)$ exactly. | $\dfrac{\sqrt{3}}{2}$ |
| 18 | Simplify $\sin(-\theta)$ and $\cos(-\theta)$. | $-\sin\theta$ and $\cos\theta$. |
| 19 | Solve $\tan x = -\sqrt{3}$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{2\pi}{3}$ or $\dfrac{5\pi}{3}$ |
| 20 | A sector has radius 9 cm and angle $\dfrac{4\pi}{9}$ rad. Find (a) the arc length, (b) the sector area, exactly. | (a) $4\pi$ cm; (b) $18\pi$ cm$^2$ |
| 21 | Find an angle between $0$ and $2\pi$ coterminal with $\dfrac{13\pi}{4}$. | $\dfrac{5\pi}{4}$ |
| 22 | Find the exact value of $\sin\!\left(\dfrac{11\pi}{6}\right) + \cos\!\left(\dfrac{5\pi}{6}\right)$. | $-\dfrac{1 + \sqrt{3}}{2}$ |
| 23 | Solve $2\sin^2 x - 1 = 0$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{4}, \dfrac{3\pi}{4}, \dfrac{5\pi}{4}, \dfrac{7\pi}{4}$ |
| 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. | $\approx 61.4$ cm$^2$ |
| 25 | Verify the identity $\dfrac{1 - \cos^2\theta}{\sin\theta} = \sin\theta$. | LHS $= \dfrac{\sin^2\theta}{\sin\theta} = \sin\theta$. |
| 26 | If $\sin\theta = -\dfrac{4}{5}$ and $\theta$ is in QIII, find $\cos\theta$ and $\tan\theta$. | $\cos\theta = -\dfrac{3}{5}$, $\tan\theta = \dfrac{4}{3}$ |
| 27 | Solve $\sin(2x) = \dfrac{1}{2}$ for $0 \leq x < 2\pi$. | $x = \dfrac{\pi}{12}, \dfrac{5\pi}{12}, \dfrac{13\pi}{12}, \dfrac{17\pi}{12}$ |
| 28 | Use $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ and a half-angle / difference identity to find $\sin\dfrac{\pi}{12}$ exactly. | $\dfrac{\sqrt{6} - \sqrt{2}}{4}$ |
| 29 | Convert $\dfrac{7\pi}{12}$ rad to degrees. | $105^\circ$ |
| 30 | Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x < 2\pi$. | $x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$ |
| 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. | $r = 12$; segment $\approx 13.7$ cm$^2$ |
| 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. | $\theta = \dfrac{3}{2}$ rad $\approx 85.9^\circ$ |
| 33 | Prove the identity $\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = \dfrac{2}{\cos\theta}$. | See working. |
| 34 | Solve $\cos 2x = \sin x$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$ |
| 35 | A sector has perimeter 20 cm. Find the radius that maximises the area. | $r = 5$ cm; max area 25 cm$^2$ |
| 36 | Find the exact value of $\sin\!\left(\dfrac{7\pi}{12}\right)$ using $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$. | $\dfrac{\sqrt{6} + \sqrt{2}}{4}$ |
| 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. | $\theta = \dfrac{\pi}{3}$; segment $= 6\pi - 9\sqrt{3} \approx 3.26$ cm$^2$ |
| 38 | Solve $\sin x + \cos x = 1$ for $0 \leq x \leq 2\pi$. | $x = 0, \dfrac{\pi}{2}, 2\pi$ |
| 39 | An annular sector has inner radius 4 cm, outer radius 7 cm, and angle $\dfrac{\pi}{3}$ rad. Find its area exactly. | $\dfrac{11\pi}{2}$ cm$^2$ |
| 40 | Show that if $\sin\theta + \cos\theta = \dfrac{1}{2}$, then $\sin\theta \cos\theta = -\dfrac{3}{8}$. | See working. |
Pack B — Answers
| # | Question | Answer |
|---|---|---|
| 1 | Convert $ 135^\circ$ to radians, as an exact multiple of $\pi$. | $\dfrac{3\pi}{4}$ |
| 2 | Convert $\dfrac{ 2\pi}{ 3}$ radians to degrees. | $120^\circ$ |
| 3 | Find the exact value of $\sin\!\left(\dfrac{\pi}{ 4}\right)$. | $\dfrac{\sqrt{2}}{2}$ |
| 4 | Find the exact value of $\cos\!\left(\dfrac{\pi}{ 6}\right)$. | $\dfrac{\sqrt{3}}{2}$ |
| 5 | Find the exact value of $\tan\!\left(\dfrac{\pi}{ 3}\right)$. | $\sqrt{3}$ |
| 6 | A sector has radius 8 cm and angle $ 1.5$ rad. Find the arc length. | 12 cm |
| 7 | A sector has radius 10 cm and angle $ 0.6$ rad. Find its area. | 30 cm$^2$ |
| 8 | A sector has radius 8 cm and angle $ 45^\circ$. Find the arc length, to 3 s.f. | 6.28 cm |
| 9 | A sector has radius 10 cm and angle $ 60^\circ$. Find the sector area, to 3 s.f. | 52.4 cm$^2$ |
| 10 | Using a 45-45-90 triangle, give the exact value of $\sin 45^\circ$ and $\cos 45^\circ$. | $\sin 60^\circ = \dfrac{\sqrt{3}}{2}$; $\cos 60^\circ = \dfrac{1}{2}$. |
| 11 | Find the exact value of $\cos\!\left(\dfrac{ 2\pi}{ 3}\right)$. | $-\dfrac{1}{2}$ |
| 12 | Find the exact value of $\sin\!\left(\dfrac{ 5\pi}{ 4}\right)$. | $-\dfrac{\sqrt{2}}{2}$ |
| 13 | Solve $2\cos x = -1$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{4}$ or $\dfrac{3\pi}{4}$ |
| 14 | Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in QI, find $\cos\theta$ exactly. | $\dfrac{12}{13}$ |
| 15 | If $\sin\theta = \dfrac{3}{5}$ and $\cos\theta = \dfrac{4}{5}$, find $\tan\theta$. | $\dfrac{5}{12}$ |
| 16 | Find the perimeter of a sector with radius 5 cm and angle $\dfrac{ 3\pi}{ 4}$ rad, to 3 s.f. | 21.8 cm |
| 17 | State the value of $\sin\!\left(\dfrac{\pi}{2} - \dfrac{\pi}{6}\right)$ exactly. | $\dfrac{\sqrt{3}}{2}$ |
| 18 | Simplify $\sin(-\theta)$ and $\cos(-\theta)$. | $-\tan\theta$; since $\tan(-\theta) = \frac{\sin(-\theta)}{\cos(-\theta)} = \frac{-\sin\theta}{\cos\theta}$. |
| 19 | Solve $\tan x = -\sqrt{3}$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{4}$ or $\dfrac{5\pi}{4}$ |
| 20 | A sector has radius 9 cm and angle $\dfrac{4\pi}{9}$ rad. Find (a) the arc length, (b) the sector area, exactly. | (a) $10\pi$ cm; (b) $60\pi$ cm$^2$ |
| 21 | Find an angle between $0$ and $2\pi$ coterminal with $\dfrac{13\pi}{4}$. | $\dfrac{5\pi}{3}$ |
| 22 | Find the exact value of $\sin\!\left(\dfrac{11\pi}{6}\right) + \cos\!\left(\dfrac{5\pi}{6}\right)$. | $-\dfrac{1 + \sqrt{3}}{2}$ |
| 23 | Solve $2\sin^2 x - 1 = 0$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$ |
| 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. | $\approx 52.8$ cm$^2$ |
| 25 | Verify the identity $\dfrac{1 - \cos^2\theta}{\sin\theta} = \sin\theta$. | LHS $= (1 - \cos^2\theta) - \cos^2\theta = 1 - 2\cos^2\theta$ = RHS. |
| 26 | If $\sin\theta = -\dfrac{4}{5}$ and $\theta$ is in QIII, find $\cos\theta$ and $\tan\theta$. | $\sin\theta = \dfrac{12}{13}$, $\tan\theta = -\dfrac{12}{5}$ |
| 27 | Solve $\sin(2x) = \dfrac{1}{2}$ for $0 \leq x < 2\pi$. | $x = \dfrac{\pi}{8}, \dfrac{7\pi}{8}, \dfrac{9\pi}{8}, \dfrac{15\pi}{8}$ |
| 28 | Use $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ and a half-angle / difference identity to find $\sin\dfrac{\pi}{12}$ exactly. | $\dfrac{\sqrt{6} + \sqrt{2}}{4}$ |
| 29 | Convert $\dfrac{7\pi}{12}$ rad to degrees. | $110^\circ$ |
| 30 | Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x < 2\pi$. | $x = 0, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}$ |
| 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. | $r \approx 13.07$; segment $\approx 4.85$ cm$^2$ |
| 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. | $\theta = 2$ rad $\approx 114.6^\circ$ |
| 33 | Prove the identity $\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = \dfrac{2}{\cos\theta}$. | See working. |
| 34 | Solve $\cos 2x = \sin x$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{\pi}{2}, \dfrac{3\pi}{2}$ |
| 35 | A sector has perimeter 20 cm. Find the radius that maximises the area. | $r = 7.5$ cm; max area 56.25 cm$^2$ |
| 36 | Find the exact value of $\sin\!\left(\dfrac{7\pi}{12}\right)$ using $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$. | $\dfrac{\sqrt{2} - \sqrt{6}}{4}$ |
| 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. | $\theta = \dfrac{\pi}{3}$; segment $= \dfrac{32\pi}{3} - 16\sqrt{3} \approx 5.79$ cm$^2$ |
| 38 | Solve $\sin x + \cos x = 1$ for $0 \leq x \leq 2\pi$. | $x = \dfrac{\pi}{4}, \dfrac{5\pi}{4}$ |
| 39 | An annular sector has inner radius 4 cm, outer radius 7 cm, and angle $\dfrac{\pi}{3}$ rad. Find its area exactly. | $2\pi$ cm$^2$ |
| 40 | Show that if $\sin\theta + \cos\theta = \dfrac{1}{2}$, then $\sin\theta \cos\theta = -\dfrac{3}{8}$. | See working: $\sin\theta \cos\theta = 0$. |
Problems — Worked Solutions
**Exact values [EXT].** State the exact value of: (a) $\sin 60^\circ$ (b) $\cos 30^\circ$ (c) $\tan 45^\circ$ (d) $\sin\dfrac{\pi}{6}$
(a) $\frac{\sqrt{3}}{2}$. (b) $\frac{\sqrt{3}}{2}$. (c) 1. (d) $\frac{1}{2}$.
**Quadrant II [EXT].** Find the exact value of $\cos\!\left(\dfrac{5\pi}{6}\right)$. Show the reference angle, the quadrant, and the sign.
$-\dfrac{\sqrt{3}}{2}$.
**Fan-blade sector [EXT].** A circular fan blade of radius 9 cm sweeps out a sector with central angle $\dfrac{4\pi}{9}$ rad. (a) Find the length of the arc swept. (b) Find the area of the swept sector. (c) Find the perimeter of the swept sector.
(a) $4\pi \approx 12.6$ cm. (b) $18\pi \approx 56.5$ cm$^2$. (c) $18 + 4\pi \approx 30.6$ cm.
**Trig equations exactly [EXT].** Solve each in the given range. (a) $2\cos x = -1$ for $0 \leq x \leq 2\pi$ (b) $\sin x = \dfrac{\sqrt{2}}{2}$ for $0^\circ \leq x \leq 360^\circ$ (c) $\tan x = -\sqrt{3}$ for $0 \leq x \leq 2\pi$
(a) $x = \frac{2\pi}{3}$ or $\frac{4\pi}{3}$. (b) $x = 45^\circ$ or $135^\circ$. (c) $x = \frac{2\pi}{3}$ or $\frac{5\pi}{3}$.
**Pythagorean identity [EXT].** Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in Quadrant II, find: (a) $\cos\theta$ (b) $\tan\theta$ (c) $\sin\theta \cos\theta$
(a) $-\frac{4}{5}$. (b) $-\frac{3}{4}$. (c) $-\frac{12}{25}$.
**Segment area [EXT].** A chord of a circle of radius 10 cm subtends an angle of $\dfrac{2\pi}{3}$ rad at the centre. (a) Find the area of the larger of the two regions (i.e. the major sector). (b) Find the area of the triangle formed by the chord and the two radii. (c) Hence find the area of the minor segment, to 3 s.f.
(a) Major sector $\frac{200\pi}{3} \approx 209.4$ cm$^2$. (b) $25\sqrt{3} \approx 43.3$ cm$^2$. (c) Minor segment $\approx 61.4$ cm$^2$.
**Identity proof [EXT].** Prove the identity $\dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}$, stating any necessary restrictions on $\theta$.
See working.
**Quadratic in $\sin$ [EXT].** Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x \leq 2\pi$.
$x = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{3\pi}{2}$.
**Sum formula [EXT].** Use $\frac{7\pi}{12} = \frac{\pi}{4} + \frac{\pi}{3}$ and the sum formula to find: (a) $\sin\!\left(\dfrac{7\pi}{12}\right)$ (b) $\cos\!\left(\dfrac{7\pi}{12}\right)$
(a) $\frac{\sqrt{6} + \sqrt{2}}{4}$. (b) $\frac{\sqrt{2} - \sqrt{6}}{4}$.
**Sector optimisation [EXT].** A sector has perimeter 24 cm. (a) Let the radius be $r$ cm. Express the central angle in radians in terms of $r$. (b) Express the area in terms of $r$. (c) Find the radius that maximises the area, and state the maximum area.
(a) $\theta = \dfrac{24 - 2r}{r}$. (b) $A = r(12 - r) = 12r - r^2$. (c) $r = 6$, max area 36 cm$^2$.
**Annular sector [EXT].** An annular ring has inner radius 4 cm, outer radius 7 cm. (a) Find the area of the full annulus. (b) An annular sector of angle $\frac{\pi}{3}$ rad is cut from the ring. Find its area, exact and to 3 s.f. (c) Find the perimeter of the annular sector (including both arcs and the two radial edges), to 3 s.f.
(a) $33\pi \approx 104$ cm$^2$. (b) $\frac{11\pi}{2} \approx 17.3$ cm$^2$. (c) $11\pi/3 + 6 \approx 17.5$ cm. Wait — recompute: outer arc $= 7 \cdot \frac{\pi}{3} = \frac{7\pi}{3}$; inner arc $= 4 \cdot \frac{\pi}{3} = \frac{4\pi}{3}$; two radial edges total $2 \cdot 3 = 6$. Total perimeter $= \frac{7\pi + 4\pi}{3} + 6 = \frac{11\pi}{3} + 6 \approx 17.5$ cm.
**Mini-investigation — equation that almost has a closed form [EXT].** Consider the equation $\sin x = \dfrac{x}{2}$ for $x \geq 0$. (a) Verify by inspection that $x = 0$ is a solution. (b) Sketch $y = \sin x$ and $y = x/2$ on the same axes for $-\pi \leq x \leq \pi$. How many other solutions can you identify? (c) Use a calculator / GDC to find the positive non-zero solution to 3 s.f. (d) Comment on why this equation has no closed-form algebraic solution.
(a) Both sides give 0. (b) Two more solutions visible, one positive, one negative. (c) $x \approx 1.90$. (d) The equation mixes a transcendental ($\sin$) and a polynomial — there is no algebraic closed-form.