Unit 2 Topic 2 Review — Applications of Trigonometry

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Review Questions

1. Q1 — Right-angle trigonometry

   Fluency

   In a right-angled triangle, one angle is 38° and the hypotenuse is 22 cm.

   (a) Find the side opposite the 38° angle.

   (b) Find the side adjacent to the 38° angle.

2. Q2 — Angle of elevation

   Fluency

   From a point 65 m from the base of a mast (on level ground), the top of the mast is observed at an angle of elevation of 42°. Find the height of the mast.

3. Q3 — Sine rule to find sides

   Fluency

   In triangle ABC: angle A = 58°, angle B = 75°, and side a = 18 cm.

   (a) Find side b.

   (b) Find side c (note: C = 180° − 58° − 75° = 47°).

4. Q4 — Cosine rule to find a side

   Fluency

   In triangle ABC: a = 11 m, b = 16 m and angle C = 64°. Find side c.

5. Q5 — Area of a triangle

   Fluency

   A triangular garden has two sides of 14 m and 20 m with an included angle of 55°. Find the area of the garden.

6. Q6 — Ladder against a wall

   Understanding

   A ladder 5.5 m long leans against a vertical wall, making an angle of 68° with the horizontal ground.

   (a) Find the height the ladder reaches on the wall.

   (b) Find the distance of the base of the ladder from the wall.

7. Q7 — Angles of depression to two boats

   Understanding

   From the top of a 120 m cliff, two boats are seen on the same side. The first boat is at an angle of depression of 28° and the second at 42°.

   (a) Find the horizontal distance to the first boat.

   (b) Find the horizontal distance to the second boat.

   (c) Find the distance between the two boats.

8. Q8 — Bearing and displacement

   Understanding

   A ship sails from port on a bearing of 125° for 75 km.

   (a) How far south of port is the ship?

   (b) How far east of port is the ship?

9. Q9 — Sine rule in surveying

   Understanding

   A surveyor stands at point C and observes two points A and B. The angle ACB = 68°, angle CAB = 54°, and the distance AB = 120 m.

   (a) Find the distance CA.

   (b) Find the distance CB.

10. Q10 — Cosine rule to find the smallest angle

    Understanding

    A triangle has sides of 9 m, 15 m and 20 m. Find the smallest angle of the triangle.

11. Q11 — Area then perimeter

    Understanding

    A triangle has two sides of 18 cm and 25 cm with an included angle of 72°.

    (a) Find the area of the triangle.

    (b) Find the length of the third side.

    (c) Find the perimeter.

12. Q12 — Triangular paddock

    Problem Solving

    A triangular paddock has two sides of 280 m and 350 m with an included angle of 62°.

    (a) Find the length of the third side.

    (b) Find all three angles of the paddock.

    (c) Find the area of the paddock.

    (d) Find the cost to fence the entire paddock at $15 per metre.

13. Q13 — Navigation problem with two legs

    Problem Solving

    A ship sails 40 km on a bearing of 050°, then turns and sails 65 km on a bearing of 140°.

    (a) Find the angle between the two legs of the journey.

    (b) Find the direct distance from the starting point to the final position.

    (c) Find the bearing to sail directly back to the starting port.

14. Q14 — Two observers and tower height

    Problem Solving

    Two observers A and B are 300 m apart on level ground, both on the same side of a vertical tower. Observer A sees the top of the tower at an elevation of 35° and observer B (closer to the tower) sees it at 52°.

    (a) Set up two equations linking tower height h and horizontal distance x from A to the base of the tower.

    (b) Find the height of the tower.

    (c) Find the horizontal distance from A to the base of the tower.

15. Q15 — Land parcel: angles, area and value

    Problem Solving

    A triangular land parcel has three sides measured as 95 m, 130 m and 160 m.

    (a) Find all three angles using the cosine rule.

    (b) Find the area of the land parcel.

    (c) If land in this area is valued at $1850 per 100 m², find the total value of the parcel.