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.
15 questions · official format, shuffled order
1 A right-angled triangle has hypotenuse 26 cm and one shorter side 10 cm. The angle θ is opposite the 10 cm side. What is cos θ?
Show worked solution
The third side is √(676 − 100) = 24, so cos θ = 24/26 = 12/13. (5/13 is sin θ.)
2 θ is an acute angle and tan θ = 11/60. What is sin θ?
Show worked solution
The hypotenuse is √(121 + 3600) = √3721 = 61, so sin θ = 11/61. (60/61 is cos θ.)
3 In a right-angled triangle, one angle is 30° and the hypotenuse is 18 cm. How long is the side opposite the 30° angle?
Show worked solution
18 × ½ = 9 cm.
4 In a right-angled triangle, an angle of 65° is opposite the side x, and the hypotenuse is 40 cm. Using sin 65° ≈ 0.9, find x.
Show worked solution
40 × 0.9 = 36 cm. (44.4 divides 40 by 0.9 instead of multiplying.)
5 An angle of 55° sits between the base and the hypotenuse. The side opposite it is 28 cm. Using tan 55° ≈ 1.4, how long is the base?
Show worked solution
28 ÷ 1.4 = 20 cm. (39.2 multiplies by 1.4 instead of dividing — the unknown is on the bottom.)
6 A right-angled triangle has shorter sides 5 cm and 5√3 cm. What is the angle opposite the 5 cm side?
Show worked solution
tan θ = 1/√3, so θ = 30°. (60° is the angle opposite the 5√3 side.)
7 An isosceles right-angled triangle has hypotenuse 14 cm. How long is each of its shorter sides?
Show worked solution
2a2 = 196, so a2 = 98 and a = 7√2 cm. (7 halves the hypotenuse, which only works opposite a 30° angle.)
8 From a point 18 m from the foot of a tower, the angle of elevation of the top is 60°. How tall is the tower?
Show worked solution
h = 18 × √3 = 18√3 m. (36 is the line of sight, 18 ÷ cos 60°, not the height.)
9 From the top of a 65 m cliff, the angle of depression of a boat is 45°. How far is the boat from the foot of the cliff?
Show worked solution
tan 45° = 1, so the distance equals the height: 65 m. (92 is the line of sight, 65√2.)
10 A tower is 33 m tall. From point P the angle of elevation of its top is 45°; from point Q, further away on the same side, it is 30°. How far apart are P and Q?
Show worked solution
P is 33 m away and Q is 33√3 m away, so PQ = 33√3 − 33 m. (33√3 is Q’s distance from the tower, not from P.)
11 What is the compass bearing S35°W written as a true bearing?
Show worked solution
180° + 35° = 215°. (145° turns towards east instead of west.)
12 The bearing of B from A is 064°. What is the bearing of A from B?
Show worked solution
64° + 180° = 244°. (116° subtracts from 180°, which is not the reverse direction.)
13 A hiker walks 20 km east, then 21 km south. How far is she from her starting point?
Show worked solution
√(400 + 441) = √841 = 29 km. (41 adds the legs; 841 forgets the square root.)
14 A ship sails 5 km north, then 5√3 km east. What is its true bearing from its starting point?
Show worked solution
tan = √3, so the angle from north is 60°: 060°. (030° measures from east instead of north.)
15 A right-angled triangle has sides 7, 24 and 25. θ is the angle opposite the side of length 7.
1 sin θ = 7/25.
2 cos θ = 24/25.
3 tan θ = 24/7.
Which statements are correct?
1 sin θ = 7/25.
2 cos θ = 24/25.
3 tan θ = 24/7.
Which statements are correct?
Show worked solution
sin θ = 7/25 (1 ✓). cos θ = 24/25 (2 ✓). tan θ = 7/24, not 24/7 (3 ✗). 1 and 2.