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IB Mathematics: Applications and Interpretation HL · 12.10: Geometry & Trigonometry · 12.10.13

Name Date

The sine rule: the ambiguous case Problem set A

Sketch each triangle. Give angles to 1 decimal place or 3 significant figures, and check the angle sum for a second solution.

Difficulty: first skills applying them stretch

Skills

  1. In triangle ABC, angle A = 72°, BC = 14.2 cm and AC = 11.5 cm. Find angle B.

    14.2 cm11.5 cm72°ABC

  2. In triangle ABC, angle A = 112°, BC = 16.5 cm and AC = 9.8 cm. Find angle B.

    16.5 cm9.8 cm112°ABC

  3. In triangle ABC, angle A = 32°, BC = 8.5 cm and AC = 12.4 cm. Find the two possible values of angle B.

    8.5 cm12.4 cm8.5 cm32°AB′BC

  4. In triangle ABC, angle A = 46°, BC = 9.2 cm and AC = 11.8 cm. Angle B can have two values. Find

    1. (a) both possible values of angle B
    2. (b) the angle C in each case

  5. In triangle ABC, angle A = 50° and AC = 10 cm. How many different triangles are there when BC is

    1. (a) a = 7 cm
    2. (b) a = 9 cm
    3. (c) a = 12 cm
    4. (d) a = 10 cm

In context

  1. A boat leaves a harbour H and sails 14 km to a buoy B. A lighthouse L is 11 km from the buoy and angle LHB = 40°. Find

    11 km14 km11 km40°HL′LB
    1. (a) the two possible sizes of angle HLB
    2. (b) the two possible distances HL

  2. Radar stations R and S track a plane P. PS = 52 km, RP = 38 km and angle PRS = 62°. Find angle PSR.

    52 km38 km62°θRSP

Stretch

  1. A designer makes a triangular frame ABC with AB = 15 cm and angle A = 36°. The side BC has length x cm. Find the values of x for which two different frames can be made.

  2. In triangle ABC, angle A = 28°, BC = 7.2 cm and AC = 13.5 cm. There are two possible triangles. Find

    1. (a) both possible values of angle B
    2. (b) the area of each triangle

jmaths.xyz · AI HL · 12.10.13 The sine rule: the ambiguous case