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IB Mathematics: Applications and Interpretation HL · 12.10: Geometry & Trigonometry · 12.10.11

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The unit circle and trigonometric equations Problem set A

Check the calculator mode first: degrees or radians. Give angles to 1 decimal place in degrees, or 3 decimal places in radians, unless told otherwise.

Difficulty: first skills applying them stretch

Skills

  1. A point P lies on a circle with centre O at the origin and radius 8 cm. The line OP makes an angle of 205° with the positive x-axis, measured anticlockwise. Find the coordinates of P.

  2. The angle θ is obtuse and \(\sin\theta=\frac{8}{17}\). Find the exact value of \(\cos\theta\).

  3. The angle θ is between 180° and 270° and \(\cos\theta=-0.6\).

    1. (a) Find \(\sin\theta\).
    2. (b) Find \(\tan\theta\).

  4. Given that sin 24° = 0.407 and cos 24° = 0.914 (3 s.f.), use the unit circle to answer the following.

    1. (a) Write down another angle between 0° and 360° with the same sine as 24°.
    2. (b) Write down another angle between 0° and 360° with the same cosine as 24°.
    3. (c) cos 0.8 = 0.697 (radians). Write down another angle between 0 and 2π with the same cosine.

  5. Solve each equation for \(0^\circ\le x\le360^\circ\).

    1. (a) \(\sin x=0.72\)
    2. (b) \(\cos x=-0.25\)

  6. Solve \(4\cos 2x=1\) for \(0\le x\le\pi\). Give the solutions in radians to 3 decimal places.

In context

  1. A Ferris wheel has radius 15 m and its lowest point is 2 m above the ground. A seat is at the lowest point when t = 0, where t is the time in minutes. One turn takes 12 minutes. The height of the seat above the ground is \(h=17-15\cos\left(\frac{\pi t}{6}\right)\) metres, for \(0\le t\le12\).

    2468101248121620242832t (min)h (m)h
    1. (a) Find the height of the seat when t = 2.
    2. (b) Find the times when the seat is 20 m above the ground.
    3. (c) Find for how long in one turn the seat is more than 20 m above the ground.

  2. A point P moves round a circle with centre the origin and radius 12 cm. At t = 0 it is at (12, 0). It moves anticlockwise through 0.4 radians every second. Use radian mode.

    1. (a) Find the coordinates of P after 5 seconds.
    2. (b) Find the first time at which the y-coordinate of P is 9 cm.

Stretch

  1. The point P(−0.6, y) lies on the unit circle, in the second quadrant. The angle θ is measured anticlockwise from the positive x-axis, with 0 ≤ θ ≤ 2π.

    1. (a) Find y.
    2. (b) Find θ in radians.
    3. (c) Find tan θ.

  2. Consider the equation \(3\sin x=2\cos x\), for \(0^\circ\le x\le360^\circ\).

    1. (a) Show that the equation can be written as \(\tan x=\frac{2}{3}\).
    2. (b) Hence solve the equation.

jmaths.xyz · AI HL · 12.10.11 The unit circle and trigonometric equations