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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 B

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

  1. A point P lies on a circle with centre O at the origin and radius 6.5 cm. The line OP makes an angle of 310° with the positive x-axis, measured anticlockwise. Find the coordinates of P.

  2. The angle θ is between 270° and 360° and \(\cos\theta=\frac{12}{37}\). Find the exact value of \(\sin\theta\).

  3. The angle θ is between 180° and 270° and \(\sin\theta=-0.28\).

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

  4. Given that sin 37° = 0.602 and cos 37° = 0.799 (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 37°.
    2. (b) Write down another angle between 0° and 360° with the same cosine as 37°.
    3. (c) cos 1.3 = 0.267 (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) \(\sin x=0.46\)
    2. (b) \(\cos x=-0.62\)

  6. Solve \(6\sin 2x=2.5\) for \(0\le x\le\pi\). Give the solutions in radians to 3 decimal places.

In context

  1. The depth of water in a harbour is \(d=6-2.4\cos\left(\frac{\pi t}{6.2}\right)\) metres, where t is the time in hours after low tide, for \(0\le t\le12.4\).

    246810122468t (h)d (m)d
    1. (a) Find the depth of water 2 hours after low tide.
    2. (b) Find the times when the depth is 7 m.
    3. (c) A ship can enter the harbour when the depth is more than 7 m. For how long in one tide cycle can it enter?

  2. The tip P of a fan blade moves round a circle with centre the origin and radius 25 cm. At t = 0 it is at (25, 0). It moves anticlockwise through 0.7 radians every second. Use radian mode.

    1. (a) Find the coordinates of P after 3 seconds.
    2. (b) Find the first time at which the y-coordinate of P is 15 cm.

Stretch

  1. The point P(0.8, y) lies on the unit circle, in the fourth 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 \(2\sin x=5\cos x\), for \(0^\circ\le x\le360^\circ\).

    1. (a) Show that the equation can be written as \(\tan x=\frac{5}{2}\).
    2. (b) Hence solve the equation.

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