Mathematics
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Fluency · Pack B

11.8 Unit Circle

Answer each question. Show working where needed.

Bronze

  1. Convert $ 135^\circ$ to radians, as an exact multiple of $\pi$.

  2. Convert $\dfrac{ 2\pi}{ 3}$ radians to degrees.

  3. Find the exact value of $\sin\!\left(\dfrac{\pi}{ 4}\right)$.

  4. Find the exact value of $\cos\!\left(\dfrac{\pi}{ 6}\right)$.

  5. Find the exact value of $\tan\!\left(\dfrac{\pi}{ 3}\right)$.

  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 $\cos 45^\circ$.

Silver

  1. Find the exact value of $\cos\!\left(\dfrac{ 2\pi}{ 3}\right)$.

  2. Find the exact value of $\sin\!\left(\dfrac{ 5\pi}{ 4}\right)$.

  3. Solve $2\cos x = -1$ for $0 \leq x \leq 2\pi$.

  4. Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in QI, find $\cos\theta$ exactly.

  5. If $\sin\theta = \dfrac{3}{5}$ and $\cos\theta = \dfrac{4}{5}$, find $\tan\theta$.

  6. Find the perimeter of a sector with radius 5 cm and angle $\dfrac{ 3\pi}{ 4}$ rad, to 3 s.f.

  7. State the value of $\sin\!\left(\dfrac{\pi}{2} - \dfrac{\pi}{6}\right)$ exactly.

  8. Simplify $\sin(-\theta)$ and $\cos(-\theta)$.

  9. Solve $\tan x = -\sqrt{3}$ for $0 \leq x \leq 2\pi$.

  10. A sector has radius 9 cm and angle $\dfrac{4\pi}{9}$ rad. Find (a) the arc length, (b) the sector area, exactly.

Gold

  1. Find an angle between $0$ and $2\pi$ coterminal with $\dfrac{13\pi}{4}$.

  2. Find the exact value of $\sin\!\left(\dfrac{11\pi}{6}\right) + \cos\!\left(\dfrac{5\pi}{6}\right)$.

  3. Solve $2\sin^2 x - 1 = 0$ for $0 \leq x \leq 2\pi$.

  4. 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.

  5. Verify the identity $\dfrac{1 - \cos^2\theta}{\sin\theta} = \sin\theta$.

  6. If $\sin\theta = -\dfrac{4}{5}$ and $\theta$ is in QIII, find $\cos\theta$ and $\tan\theta$.

  7. Solve $\sin(2x) = \dfrac{1}{2}$ for $0 \leq x < 2\pi$.

  8. Use $\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$ and a half-angle / difference identity to find $\sin\dfrac{\pi}{12}$ exactly.

  9. Convert $\dfrac{7\pi}{12}$ rad to degrees.

  10. Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x < 2\pi$.

Platinum

  1. 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.

  2. 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.

  3. Prove the identity $\dfrac{1 + \sin\theta}{\cos\theta} + \dfrac{\cos\theta}{1 + \sin\theta} = \dfrac{2}{\cos\theta}$.

  4. Solve $\cos 2x = \sin x$ for $0 \leq x \leq 2\pi$.

  5. A sector has perimeter 20 cm. Find the radius that maximises the area.

  6. Find the exact value of $\sin\!\left(\dfrac{7\pi}{12}\right)$ using $\frac{7\pi}{12} = \frac{\pi}{3} + \frac{\pi}{4}$.

  7. 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.

  8. Solve $\sin x + \cos x = 1$ for $0 \leq x \leq 2\pi$.

  9. An annular sector has inner radius 4 cm, outer radius 7 cm, and angle $\dfrac{\pi}{3}$ rad. Find its area exactly.

  10. Show that if $\sin\theta + \cos\theta = \dfrac{1}{2}$, then $\sin\theta \cos\theta = -\dfrac{3}{8}$.