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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.8 Unit Circle

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Exact values [EXT]. State the exact value of:

(a) sin⁡60∘\sin 60^\circ
(b) cos⁡30∘\cos 30^\circ
(c) tan⁡45∘\tan 45^\circ
(d) sin⁡π6\sin\dfrac{\pi}{6}

Working space

2Problem 2 of 12
Quadrant II [EXT]. Find the exact value of cos⁡ ⁣(5π6)\cos\!\left(\dfrac{5\pi}{6}\right). Show the reference angle, the quadrant, and the sign.

Working space

3Problem 3 of 12
Fan-blade sector [EXT]. A circular fan blade of radius 9 cm sweeps out a sector with central angle 4π9\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.

Working space

4Problem 4 of 12
Trig equations exactly [EXT]. Solve each in the given range.

(a) 2cos⁡x=−12\cos x = -1 for 0≤x≤2π0 \leq x \leq 2\pi
(b) sin⁡x=22\sin x = \dfrac{\sqrt{2}}{2} for 0∘≤x≤360∘0^\circ \leq x \leq 360^\circ
(c) tan⁡x=−3\tan x = -\sqrt{3} for 0≤x≤2π0 \leq x \leq 2\pi

Working space

5Problem 5 of 12
Pythagorean identity [EXT]. Given sin⁡θ=35\sin\theta = \dfrac{3}{5} and θ\theta is in Quadrant II, find:

(a) cos⁡θ\cos\theta
(b) tan⁡θ\tan\theta
(c) sin⁡θcos⁡θ\sin\theta \cos\theta

Working space

6Problem 6 of 12
Segment area [EXT]. A chord of a circle of radius 10 cm subtends an angle of 2π3\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.

Working space

7Problem 7 of 12
Identity proof [EXT]. Prove the identity 1−cos⁡θsin⁡θ=sin⁡θ1+cos⁡θ\dfrac{1 - \cos\theta}{\sin\theta} = \dfrac{\sin\theta}{1 + \cos\theta}, stating any necessary restrictions on θ\theta.

Working space

8Problem 8 of 12
Quadratic in sin⁡\sin [EXT]. Solve 2sin⁡2x+sin⁡x−1=02\sin^2 x + \sin x - 1 = 0 for 0≤x≤2π0 \leq x \leq 2\pi.

Working space

9Problem 9 of 12
Sum formula [EXT]. Use 7π12=π4+π3\frac{7\pi}{12} = \frac{\pi}{4} + \frac{\pi}{3} and the sum formula to find:

(a) sin⁡ ⁣(7π12)\sin\!\left(\dfrac{7\pi}{12}\right)
(b) cos⁡ ⁣(7π12)\cos\!\left(\dfrac{7\pi}{12}\right)

Working space

10Problem 10 of 12
Sector optimisation [EXT]. A sector has perimeter 24 cm.

(a) Let the radius be rr cm. Express the central angle in radians in terms of rr.
(b) Express the area in terms of rr.
(c) Find the radius that maximises the area, and state the maximum area.

Working space

11Problem 11 of 12
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 π3\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.

Working space

12Problem 12 of 12
Mini-investigation — equation that almost has a closed form [EXT]. Consider the equation sin⁡x=x2\sin x = \dfrac{x}{2} for x≥0x \geq 0.

(a) Verify by inspection that x=0x = 0 is a solution.
(b) Sketch y=sin⁡xy = \sin x and y=x/2y = x/2 on the same axes for −π≤x≤π-\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.

Working space