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 Which equation gives x?
Show worked solution
15 is the hypotenuse and x is adjacent, so cos 34° = x ÷ 15 and x = 15 cos 34°. x = 15 sin 34° finds the opposite side; x = 15 tan 34° treats 15 as adjacent; x = 15 ÷ cos 34° divides though x is on top, and would make x longer than the hypotenuse.
2 Find the two shorter sides, leaving √3 in your answer.
Show worked solution
y = 26 × sin 30° = 26 × 1/2 = 13 cm. x = 26 × cos 30° = 26 × √3/2 = 13√3 cm. Check: the sides are 13 : 13√3 : 26 = 1 : √3 : 2 ✓.
3 A ramp 50 m long rises at 28° to the horizontal. sin 28° ≈ 0.47 and cos 28° ≈ 0.88. How high, in m, does the ramp rise?
Show worked solution
The height is opposite 28° and the ramp is the hypotenuse, so sin 28° = h ÷ 50 and h = 50 × 0.47 = 23.5 m. Using cos gives 44 m, which is the horizontal distance, not the height.
Level 2 · Core
4 An isosceles right-angled triangle has a hypotenuse of 10 cm. What is its area, in cm2?
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Each short side is 10 × 1/√2 = 10 ÷ √2 = 5√2 (since 5√2 × √2 = 10). Area = ½ × 5√2 × 5√2 = ½ × 50 = 25 cm2. 12.5 takes each short side as 5, half the hypotenuse; 50 forgets the ½; 25√2 uses the hypotenuse as the base, ½ × 10 × 5√2.
5 Find x and the hypotenuse h, leaving √3 in your answers.
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x = 15 ÷ tan 60° = 15 ÷ √3. Since 5√3 × √3 = 15, x = 5√3 cm. In a 30°-60°-90° triangle the hypotenuse is twice the side next to 60°, so h = 10√3 cm. Check: sin 60° = 15 ÷ 10√3 = 15√3 ÷ 30 = √3/2 ✓.
6 cos 25° ≈ 0.9 and sin 25° ≈ 0.42. Find the length x, in cm.
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27 is adjacent and x is the hypotenuse, so cos 25° = 27 ÷ x. x is on the bottom, so x = 27 ÷ 0.9 = 30 cm. Check: 30 is longer than 27 ✓. 27 × 0.9 = 24.3 is the multiply slip.
Level 3 · Challenge
7 sin 36° ≈ 0.59, cos 36° ≈ 0.81 and tan 36° ≈ 0.73.Three statements:
1 The side opposite the 36° angle is about 11.8 cm.
2 The side next to the 36° angle is about 20 ÷ 0.81 cm.
3 The opposite side divided by the adjacent side is about 0.73.
Which statements are correct?
1 The side opposite the 36° angle is about 11.8 cm.
2 The side next to the 36° angle is about 20 ÷ 0.81 cm.
3 The opposite side divided by the adjacent side is about 0.73.
Which statements are correct?
Show worked solution
Opposite = 20 × 0.59 = 11.8 (1 ✓). Adjacent = 20 × cos 36° = 20 × 0.81 = 16.2; dividing gives about 24.7, longer than the hypotenuse (2 ✗). 11.8 ÷ 16.2 ≈ 0.73, which is tan 36° (3 ✓). 1 and 3.
8 A diagonal of a rectangle makes an angle of 30° with the longer side. The shorter side is 7 cm. Find the longer side and the diagonal.
Show worked solution
tan 30° = 7 ÷ L, so L = 7 ÷ (1/√3) = 7√3 cm. sin 30° = 7 ÷ d, so d = 7 ÷ 1/2 = 14 cm. Check: the sides are 7 : 7√3 : 14 = 1 : √3 : 2 ✓.
9 tan 31° ≈ 0.6. The right angle is at C, D lies on BC, AC = 12 cm and angle ADC = 45°. How long, in cm, is BD?
Show worked solution
In triangle ABC, tan 31° = 12 ÷ BC, so BC = 12 ÷ 0.6 = 20. In triangle ADC the angle is 45°, so DC = AC = 12. BD = BC − DC = 20 − 12 = 8 cm.