Topic Review — Trigonometry

Mixed questions covering all four lessons. Click each answer button to reveal the solution.

1. Right-triangle sides and angles. Fluency

   • (a) θ = 42°, hypotenuse = 18 m. Find the side opposite θ.
   • (b) Adjacent = 9 cm, hypotenuse = 15 cm. Find θ.
   • (c) θ = 60°, opposite = 7 m. Find the hypotenuse exactly.
   • (d) A right triangle has legs 9 and 40. Find the hypotenuse and both acute angles.

2. Angles of elevation and depression. Fluency

   • (a) Elevation 35° from 50 m away. Find the height above eye level.
   • (b) A bird on a 30 m cliff looks down at a fish. Depression angle 28°. How far is the fish from the cliff base?
   • (c) Observer 1.7 m tall looks up at a treetop 15 m away at elevation 52°. Find the total tree height.
   • (d) A kite is 80 m above the ground and the string makes 40° with the ground. Find the string length.

3. Bearings. Fluency

   • (a) A ship sails 40 km on bearing 125°. Find its north and east displacement.
   • (b) State the back-bearing of 285°.
   • (c) Convert S35°W to a three-figure bearing.
   • (d) A hiker walks 6 km north then 8 km east. Find the bearing of their finish from their start.

4. Sine rule and cosine rule. Fluency

   • (a) Triangle: A=42°, B=73°, a=11 cm. Find b.
   • (b) Triangle: a=8, b=11, c=14. Find angle C.
   • (c) Triangle: two sides 10 and 13 with included angle 65°. Find the third side.
   • (d) Find the area of the triangle in (c).

5. Trigonometry from a diagram. Understanding

   The diagram shows a triangle with all three sides labelled. Use it to answer the questions.

   • (a) Which angle is largest? Explain without calculating.
   • (b) Find angle A using the Cosine Rule.
   • (c) Find angle B.
   • (d) Find the area of the triangle.

6. Two observers, one landmark. Understanding

   Two observers A and B are 200 m apart on flat ground. They both observe a tower between them. The angle of elevation from A is 32° and from B is 44°.

   • (a) Draw a diagram showing the tower, the two observers, and the angles.
   • (b) If d is the horizontal distance from A to the tower base, write two equations for the tower height h.
   • (c) Solve for d and then find h.
   • (d) Verify using the second equation.

7. Triangular field. Understanding

   A triangular field has sides 85 m, 110 m, and 140 m.

   • (a) Find the largest angle (opposite the 140 m side).
   • (b) Find the smallest angle.
   • (c) Find the area of the field.
   • (d) A fence is to be built across the field from the vertex of the largest angle, perpendicular to the longest side. Find the length of this fence.

8. Search pattern. Understanding

   A rescue plane starts at base (B), flies 120 km on bearing 045° to point P, then flies to point Q which is 90 km from B on bearing 150°.

   • (a) Find the angle PBQ.
   • (b) Find the distance PQ.
   • (c) Find the bearing of Q from P.
   • (d) Total distance flown B→P→Q.

9. Flagpole on a hill. Problem Solving

   A flagpole stands at the top of a hill. From a point A at the base of the hill, the angle of elevation of the bottom of the pole is 18° and the angle of elevation of the top of the pole is 27°. The horizontal distance from A to the base of the hill is 80 m.

   • (a) Find the height of the hilltop above A’s level.
   • (b) Find the height of the top of the flagpole above A’s level.
   • (c) Find the height of the flagpole itself.
   • (d) If a storm causes the pole to lean so its top is directly above point A, what would the angle of depression from the top of the pole to A now be?

10. Orienteering course. Problem Solving

    An orienteering course has checkpoints at A, B, and C. B is 3.5 km from A on bearing 072°. C is 4.8 km from A on bearing 148°. D is the midpoint of BC.

    • (a) Find the angle BAC.
    • (b) Find the distance BC.
    • (c) Find the distance AD (A to the midpoint of BC).
    • (d) Find the bearing of B from C.

11. Sine rule ambiguous case. Problem Solving

    In a triangle, a = 7, b = 10, and A = 35°.

    • (a) Use the Sine Rule to find sin B.
    • (b) Find both possible values of angle B.
    • (c) Find both possible values of side c.
    • (d) Why does this situation have two solutions?

12. Exact values and proof. Problem Solving

    • (a) Show that sin² 45° + cos² 45° = 1 using exact values.
    • (b) Find the exact perimeter of a right triangle with hypotenuse 10 and one angle 30°.
    • (c) In triangle ABC with a = b = 6 and C = 120°, find side c exactly.
    • (d) Find the exact area of the triangle in (c).