Practice Maths

Topic Review — Trigonometry

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This review covers all lessons in Trigonometry: the sine rule (including the ambiguous case), the cosine rule and area formula, and 2D and 3D trigonometry applications. Questions are mixed in difficulty.

Review Questions

  1. In triangle ABC, A = 42°, B = 67°, and a = 11 cm. Find the length of side b.
  2. In triangle ABC, a = 8, b = 12, and A = 38°. Find all possible values of angle B, checking both solutions for validity.
  3. In triangle XYZ, X = 105°, x = 20 m, and y = 9 m. Explain why there is only one possible triangle and find angle Y.
  4. Find the area of triangle PQR where PQ = 14 m, PR = 10 m, and angle QPR = 72°.
  5. In triangle ABC, the area is 30 cm², side a = 9 cm, and side b = 8 cm. Find all possible values of angle C, and for each valid case find side c.
  6. In triangle ABC, a = 7 cm, b = 9 cm, and C = 65°. Find side c using the cosine rule.
  7. In triangle ABC, a = 10, b = 7, c = 6. Find the largest angle.
  8. Two ships leave port at the same time. Ship A travels 24 km on a bearing of 050°. Ship B travels 18 km on a bearing of 140°. Find the distance between the two ships.
  9. Three towns A, B, C form a triangle. AB = 60 km, BC = 45 km, AC = 50 km. Find the angle at B (angle ABC).
  10. A triangular garden has sides of length 12 m, 15 m, and 20 m. Find the area of the garden using Heron’s formula or by first finding an angle.
  11. From the top of a 60 m cliff, the angle of depression to a boat is 28°. Find the horizontal distance from the cliff base to the boat, and the direct distance from the top of the cliff to the boat.
  12. A rectangular box has length 10 m, width 8 m, and height 5 m. Find the angle the space diagonal makes with the base of the box.
  13. A yacht sails 20 km on a bearing of 025° from port P to point A, then 15 km on a bearing of 115° from A to point B. Find the distance PB and the bearing of B from P.
  14. A right pyramid has a rectangular base 8 m × 6 m and a vertical height of 10 m. Find the slant height from the apex to the midpoint of the longer base edge, and the angle this slant makes with the base.
  15. Three surveying stations A, B, and C lie in a horizontal plane. From A, the bearing to B is 070° and the distance AB = 200 m. From A, the bearing to C is 140° and AC = 160 m. A vertical mast of height 45 m stands at B. Find the area of triangle ABC and the angle of elevation of the top of the mast from C.
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