Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Ann stands 12 m from the foot of a pole on level ground. The angle of elevation of its top is 60°. How tall is the pole, in metres?
Show worked solution
tan 60° = h ÷ 12, so h = 12 × √3 = 12√3 m. 4√3 = 12 ÷ √3 divides though h is on top; 6 = 12 × cos 60° uses the wrong ratio; 24 = 12 ÷ cos 60° is the line of sight.
2 From the top of a 42 m cliff, the angle of depression of a boat is 40°. Given tan 40° ≈ 0.84, find the angle of elevation of the cliff top from the boat, and the boat's distance from the foot of the cliff.
Show worked solution
Alternate angles: the elevation at the boat is 40°. tan 40° = 42 ÷ d, so d = 42 ÷ 0.84 = 50 m.
3 Sam's eye is 1.6 m above level ground, 30 m from a building. The angle of elevation of the top of the building is 50°. Given tan 50° ≈ 1.2, how tall is the building, in metres?
Show worked solution
Above eye level: 30 × tan 50° = 30 × 1.2 = 36 m. Add the eye height: 36 + 1.6 = 37.6 m.
Level 2 · Core
4 A tower has height h. From A the angle of elevation of its top is 45°. From B, further away on the same side, it is 30°. Which expression gives the distance AB?
Show worked solution
A is h ÷ tan 45° = h from the tower, and B is h ÷ tan 30° = h ÷ (1/√3) = h√3 from it. Same side, so AB = h√3 − h = h(√3 − 1). h and h√3 are the separate distances; h(√3 + 1) adds them, as for opposite sides.
5 From the top of a 45 m lighthouse, two boats on the same side have angles of depression of 32° and 20°. Given tan 32° ≈ 0.625 and tan 20° ≈ 0.36, how far is each boat from the foot of the lighthouse?
Show worked solution
Nearer boat (bigger angle): 45 ÷ 0.625 = 72 m. Further boat: 45 ÷ 0.36 = 125 m. (They are 53 m apart.)
6 A 24 m tower stands between A and B on level ground. The angle of elevation of its top is 39° from A and 50° from B. Given tan 39° ≈ 0.8 and tan 50° ≈ 1.2, how far apart are A and B, in metres?
Show worked solution
From A: 24 ÷ 0.8 = 30 m. From B: 24 ÷ 1.2 = 20 m. Opposite sides, so AB = 30 + 20 = 50 m.
Level 3 · Challenge
7 Three statements:
1 If the angle of depression from a balcony to a car is 35°, then the angle of elevation of the balcony from the car is 55°.
2 A 10 m pole on level ground casts a shadow 10√3 m long when the sun's angle of elevation is 30°.
3 From the top of a 15 m building, the angle of depression of a point on the ground 15 m from its base is 45°.
Which statements are correct?
1 If the angle of depression from a balcony to a car is 35°, then the angle of elevation of the balcony from the car is 55°.
2 A 10 m pole on level ground casts a shadow 10√3 m long when the sun's angle of elevation is 30°.
3 From the top of a 15 m building, the angle of depression of a point on the ground 15 m from its base is 45°.
Which statements are correct?
Show worked solution
The horizontals are parallel, so the angles are alternate and equal: 35°, not 55° (1 ✗). tan 30° = 10 ÷ shadow, so shadow = 10 ÷ (1/√3) = 10√3 m (2 ✓). tan θ = 15 ÷ 15 = 1, so θ = 45° (3 ✓). 2 and 3.
8 A flagpole stands on the edge of a building's roof. From P, 15√3 m from the building on level ground, the angle of elevation of the bottom of the flagpole is 30° and of its top is 60°. Find the height of the building and the length of the flagpole.
Show worked solution
Building: 15√3 × tan 30° = 15√3 × 1/√3 = 15 m. Top of flagpole: 15√3 × tan 60° = 15√3 × √3 = 45 m. Flagpole = 45 − 15 = 30 m.
9 A and B are on the same side of a tower, with AB = 40 m. The tangent of the angle of elevation of the top is 0.75 from A and 0.5 from B. How tall is the tower, in metres?
Show worked solution
A is h ÷ 0.75 = 4h/3 from the tower and B is h ÷ 0.5 = 2h from it. Same side: 2h − 4h/3 = 40, so 2h/3 = 40 and h = 60 m. Check: A is 80 m away and B is 120 m away, 40 m apart ✓.