11.8 Unit Circle

**Exact values [EXT].** State the exact value of: (a) $\sin 60^\circ$ (b) $\cos 30^\circ$ (c) $\tan 45^\circ$ (d) $\sin\dfrac{\pi}{6}$

**Quadrant II [EXT].** Find the exact value of $\cos\!\left(\dfrac{5\pi}{6}\right)$. Show the reference angle, the quadrant, and the sign.

**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.

**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 $0 \leq x \leq 2\pi$

**Pythagorean identity [EXT].** Given $\sin\theta = \dfrac{3}{5}$ and $\theta$ is in Quadrant II, find: (a) $\cos\theta$ (b) $\tan\theta$ (c) $\sin\theta \cos\theta$

**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.

**Identity proof [EXT].** Prove the identity $\dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}$, stating any necessary restrictions on $\theta$.

**Quadratic in $\sin$ [EXT].** Solve $2\sin^2 x + \sin x - 1 = 0$ for $0 \leq x \leq 2\pi$.

**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) $\cos\!\left(\dfrac{7\pi}{12}\right)$

**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.

**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.

**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.