Trigonometry — Ryan (1.5h)

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Instructions: This worksheet has 3 sections. Answer all questions. Section A: Foundation (15 questions). Section B: Core Application (15 questions). Section C: Challenge (6 questions). Show all working. Use a calculator. Total: 36 questions.

Key Concepts Covered

SOH CAH TOA — sin, cos, tan ratios
Finding unknown sides and angles in right-angled triangles
Angles of elevation and depression
Application to real-world problems (ladders, shadows, bearings, surveying)
3D trigonometry and combined Pythagoras + trigonometry

Section A

Foundation

~25 min

Use a calculator. Give your answers correct to 2 decimal places unless stated otherwise.

Find the angle θ.

Find the angle A.

Find the angle B.

Find the length of side x.

Find the length of side y.

Find the length of side z.

Find the length of side a.

Find the length of side b.

Find the length of side c.

Find the angle θ.

Find the length of side d.

Find the angle φ.

Find the length of side e.

Find the length of side f.

Section B

Core Application

~45 min

Show all your working. Give answers correct to 2 decimal places unless stated otherwise.

A ladder 14.2 m long leans against a vertical wall. The foot of the ladder is 11.5 m from the wall. Find the angle θ the ladder makes with the ground.

A tree is 22 m tall. A person stands 35 m from the base of the tree. Find the angle of elevation α from the person to the top of the tree.

A house has a symmetrical roof. The roof ridge is 4.5 m above the roof base, and the house is 12 m wide. Find the angle θ the roof makes with the horizontal.

A person 1.7 m tall stands in sunlight. The sun's rays make an angle of 38° with the ground. Find the length of the person's shadow s.

A kite is flown on a string 65 m long. The string makes an angle of 55° with the ground. Find the height h of the kite above the ground.

A road slopes upwards at an angle of 8.5° to the horizontal. A car travels 250 m along the road. What vertical height does the car gain?

A surveyor stands on one bank of a river and measures the angle to a tree on the opposite bank as 42° from the horizontal. The tree is 3.2 m above water level and the surveyor's eye level is 1.6 m. Find the width of the river d.

In a right-angled triangle, sin θ = 0.63 and the hypotenuse is 18.4 cm.

(a) Find the angle θ.
(b) Find the length of the opposite side.
(c) Find the length of the adjacent side.

Find the area of this triangle.

A ship sails 18 km on a bearing of 045° and then 24 km due east.

(a) How far north of its starting point is the ship?
(b) How far east of its starting point is the ship?
(c) What is the straight-line distance from the starting point?

From the top of a 45 m cliff, the angle of depression to a boat in the sea is 28°. Find the horizontal distance d from the base of the cliff to the boat.

A cable stays a vertical pole 8.5 m tall. The cable is attached to the pole 1.2 m below the top and anchored to the ground. The cable makes an angle of 62° with the ground.

(a) How long is the cable?
(b) How far from the base of the pole is the anchor point?
(c) A second cable is attached at the very top and anchored 6 m from the base. What angle does this cable make with the ground?

A flagpole stands on horizontal ground. From a point 24 m from the base, the angle of elevation to the top is 35°. Find the height h of the flagpole.

A support wire is attached to a tower 10 m tall and anchored to the ground. The wire is 12.5 m long. Find the horizontal distance d from the base of the tower to the anchor point.

A box is pushed 3.6 m up a ramp inclined at 18° to the horizontal.

(a) How high does the box rise vertically?
(b) What is the horizontal distance covered?

Section C

Challenge

~30 min

These questions require multiple steps. Show all working clearly.

From point A, the angle of elevation to the top of a tower is 30°. From point B, which is 40 m closer to the tower, the angle of elevation is 52°. Find the height of the tower.

A square-based pyramid has a base of 10 m by 10 m and a vertical height of 12 m. Find (a) the slant height of a triangular face and (b) the angle the slant edge makes with the base.

A surveyor measures the angle of elevation to a window on a building as 25°. After walking 30 m closer to the building, the angle of elevation is 40°. Both measurements are taken from eye level 1.5 m above the ground. Find the height of the window above the ground.

A tent pole is supported by two ropes anchored to the ground on opposite sides. The ropes make angles of 40° and 55° with the ground. The total distance between the anchor points is 18 m. Find the height h of the pole.

A room is a rectangular cuboid 8 m long, 6 m wide, and 5 m high. Find the length of the space diagonal (from one bottom corner to the opposite top corner).

From a point 42 m from the base of a building, the angle of elevation to the top is 32°. A guy wire runs from the top of the building to a point 15 m further out from the base. Find (a) the height of the building and (b) the length of the guy wire.

Answer Key

Q1: cos θ = 8.4/15.2 → θ = cos⁻¹(8.4/15.2) = 56.45°

Q2: tan θ = 7.3/12.5 → θ = tan⁻¹(7.3/12.5) = 30.28°

Q3: cos A = 6.2/9.7 → A = cos⁻¹(6.2/9.7) = 50.27°

Q4: sin B = 5.8/11.3 → B = sin⁻¹(5.8/11.3) = 30.88°

Q5: cos 35° = 14.2/x → x = 14.2/cos 35° = 17.33 cm

Q6: tan 48° = y/9.6 → y = 9.6 × tan 48° = 10.66 cm

Q7: tan 22° = 4.5/z → z = 4.5/tan 22° = 11.14 cm

Q8: sin 63° = a/17.8 → a = 17.8 × sin 63° = 15.86 cm

Q9: cos 15° = b/23.1 → b = 23.1 × cos 15° = 22.31 cm

Q10: sin 52° = 11.4/c → c = 11.4/sin 52° = 14.47 cm

Q11: sin θ = 9.1/16.5 → θ = sin⁻¹(9.1/16.5) = 33.47°

Q12: tan 37° = d/10.3 → d = 10.3 × tan 37° = 7.76 cm

Q13: tan φ = 3.8/5.5 → φ = tan⁻¹(3.8/5.5) = 34.64°

Q14: cos 28° = e/13.7 → e = 13.7 × cos 28° = 12.10 cm

Q15: sin 41° = 7.6/f → f = 7.6/sin 41° = 11.58 cm

Q16: cos θ = 11.5/14.2 → θ = cos⁻¹(11.5/14.2) = 35.92°

Q17: tan α = 22/35 → α = tan⁻¹(22/35) = 32.15°

Q18: Half-width = 6 m. tan θ = 4.5/6 → θ = tan⁻¹(4.5/6) = 36.87°

Q19: tan 38° = 1.7/s → s = 1.7/tan 38° = 2.18 m

Q20: sin 55° = h/65 → h = 65 × sin 55° = 53.24 m

Q21: sin 8.5° = h/250 → h = 250 × sin 8.5° = 36.95 m

Q22: Height diff = 3.2 − 1.6 = 1.6 m. tan 42° = 1.6/d → d = 1.6/tan 42° = 1.78 m

Q23(a): θ = sin⁻¹(0.63) = 39.06°

Q23(b): opp = 18.4 × 0.63 = 11.59 cm

Q23(c): adj = √(18.4² − 11.59²) = 14.19 cm

Q24: opp = 22.3 × sin 57° = 18.69 cm. Area = ½ × 14.6 × 18.69 = 136.53 cm²

Q25(a): 18 × cos 45° = 12.73 km north

Q25(b): 18 × sin 45° + 24 = 36.73 km east

Q25(c): √(12.73² + 36.73²) = 38.87 km

Q26: tan 28° = 45/d → d = 45/tan 28° = 84.63 m

Q27(a): Pole height = 8.5 − 1.2 = 7.3 m. sin 62° = 7.3/cable → cable = 8.27 m

Q27(b): tan 62° = 7.3/d → d = 7.3/tan 62° = 3.88 m

Q27(c): tan θ = 8.5/6 → θ = 54.78°

Q28: tan 35° = h/24 → h = 24 × tan 35° = 16.80 m

Q29: d = √(12.5² − 10²) = √(156.25 − 100) = √56.25 = 7.50 m

Q30(a): h = 3.6 × sin 18° = 1.11 m

Q30(b): d = 3.6 × cos 18° = 3.42 m

Q31: Let B to tower = x. tan 52° = h/x, tan 30° = h/(x+40). Solving: x = 32.87 m. h = 32.87 × tan 52° = 42.07 m

Q32(a): Slant height = √(12² + 5²) = √169 = 13.00 m

Q32(b): Half-diagonal = √(5²+5²) = 7.07 m. tan θ = 12/7.07 → θ = 59.49°

Q33: Let 2nd position distance = x. tan 40° = h/x, tan 25° = h/(x+30). Solving: x = 37.53 m. h above eye = 31.49 m. Window = 31.49 + 1.5 = 32.99 m

Q34: Let left distance = x. tan 40° = h/x, tan 55° = h/(18−x). Solving: x = 11.34 m. h = 11.34 × tan 40° = 9.51 m

Q35: Floor diagonal = √(8² + 6²) = 10 m. Space diagonal = √(10² + 5²) = √125 = 11.18 m

Q36(a): tan 32° = h/42 → h = 42 × tan 32° = 26.24 m

Q36(b): Total horizontal = 42 + 15 = 57 m. Wire = √(26.24² + 57²) = 62.75 m