This set has twelve original questions, ordered from easier to harder, covering all five lessons in right-triangle trigonometry. Questions 1 to 3 are warm-ups, 4 to 8 build accuracy, and 9 to 12 mix the skills. Give lengths to 3 significant figures and angles to 1 decimal place unless told otherwise.
Use a calculator in degree mode, and draw a triangle for every question. Attempt each on paper, write your working as in an exam, then open the answer. Use the routing list at the end for anything you missed, and try the triangle and bearings reasoning board or the mistake log and retest queue for follow-up.
Questions
1. A right-angled triangle has sides 5 cm, 12 cm and 13 cm. Angle θ is opposite the 5 cm side. Write sin θ, cos θ and tan θ as fractions.
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The hypotenuse is the longest side, 13 cm. Opposite θ is 5 cm, so adjacent is 12 cm.
sin θ = O/H = 5/13, cos θ = A/H = 12/13, tan θ = O/A = 5/12.
2. In a right-angled triangle, an angle of 54° has a hypotenuse of 6.5 cm. Find the side opposite the 54° angle.
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Opposite and hypotenuse means sine: sin 54° = x/6.5.
x = 6.5 × sin 54° = 6.5 × 0.8090 = 5.2586…, so 5.26 cm.
Check: 5.26 is less than 6.5. ✓
3. An angle of 37° has a hypotenuse of 8.2 cm. Find the adjacent side.
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Adjacent and hypotenuse means cosine: cos 37° = x/8.2.
x = 8.2 × 0.7986 = 6.5488…, so 6.55 cm.
Check: less than 8.2. ✓
4. An angle of 63° has an adjacent side of 5 cm. Find the opposite side.
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Opposite and adjacent means tangent: tan 63° = x/5.
x = 5 × tan 63° = 5 × 1.9626 = 9.813…, so 9.81 cm.
Check: a 63° angle is steeper than 45°, so the opposite side should be longer than the adjacent side. 9.81 is greater than 5. ✓
5. An angle of 48° has an opposite side of 11 cm. Find the hypotenuse.
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Opposite and hypotenuse means sine: sin 48° = 11/H.
H = 11 ÷ sin 48° = 11 ÷ 0.7431 = 14.80…, so 14.8 cm.
Check: the hypotenuse is longer than the 11 cm leg. ✓
6. An angle of 35° has an opposite side of 13 cm. Find the adjacent side.
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Opposite and adjacent means tangent: tan 35° = 13/x.
x = 13 ÷ tan 35° = 13 ÷ 0.7002 = 18.57…, so 18.6 cm.
Check: a 35° angle is under 45°, so the adjacent side should be longer than the opposite side. 18.6 is greater than 13. ✓
7. The opposite side is 5 cm and the adjacent side is 8 cm. Find the angle θ.
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tan θ = 5/8 = 0.625.
θ = tan⁻¹(0.625) = 32.005…, so 32.0°.
Check: the opposite is shorter than the adjacent, so the angle is under 45°. ✓
8. The adjacent side is 6 cm and the hypotenuse is 11 cm. Find the angle θ.
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cos θ = 6/11.
θ = cos⁻¹(6/11) = 56.94…, so 56.9°.
Check: 6 is a little over half of 11, and cos 60° = 0.5, so the angle should be a little under 60°. ✓
9. A rectangle is 15 cm long and 8 cm wide. Find the length of its diagonal, then the angle between the diagonal and the 15 cm side.
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The diagonal is the hypotenuse: d² = 15² + 8² = 225 + 64 = 289, so d = 17 cm.
The angle θ is opposite the 8 cm side and adjacent to the 15 cm side: tan θ = 8/15, so θ = tan⁻¹(8/15) = 28.07…, so 28.1°.
Check: sin θ = 8/17 gives sin⁻¹(0.4706) = 28.1°. ✓
10. A right-angled triangle has legs of 9 cm and 12 cm. Find the hypotenuse and the smaller angle.
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Hypotenuse: √(81 + 144) = √225 = 15 cm.
The smaller angle is opposite the shorter leg (9 cm): tan θ = 9/12 = 0.75, so θ = 36.9°.
Check: sin θ = 9/15 = 0.6 and sin⁻¹(0.6) = 36.9°. ✓
11. A student stands 40 m from the foot of a tower. Her eye is 1.6 m above level ground and the angle of elevation of the tower’s summit is 28°. Find the height of the tower.
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Height above eye level: tan 28° = h/40, so h = 40 × 0.5317 = 21.27… m.
Add the eye height: 21.27 + 1.6 = 22.87… m, so the tower is 22.9 m.
Check: using eye level only would give 21.3 m, which misses the 1.6 m.
12. A 5 m ladder leans against a vertical wall. Its foot is 1.4 m from the wall. Find how high it reaches, and the angle it makes with the ground.
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Height: h² = 5² − 1.4² = 25 − 1.96 = 23.04, so h = √23.04 = 4.8 m.
Angle with the ground: cos θ = 1.4/5 = 0.28, so θ = cos⁻¹(0.28) = 73.74…, so 73.7°.
Check: sin 73.7° = 0.96, and 4.8 ÷ 5 = 0.96. ✓
If you got these wrong
| What went wrong | Go back to |
|---|---|
| Picked the wrong ratio, or mixed up opposite and adjacent (questions 1 to 6) | Choose sine, cosine or tangent from named sides |
| Angle answers that are tiny, or in radians (questions 7, 8) | Find a missing angle with the correct mode |
| Forgot Pythagoras, or rounded too early (questions 9, 10, 12) | Combine Pythagoras with trigonometry |
| Missed the eye height, or drew the triangle wrongly (question 11) | Interpret an angle of elevation diagram |
| Gave an answer that is longer than the hypotenuse, or did not check | Check a triangle solution against its geometry |
If a whole group of questions felt shaky, redo the lesson named in the table before retrying a fresh question a few days later.
Questions 4 and 6 show why trigonometry needs diagrams: both use tangent, but the unknown sits in a different place. A teacher in online one-to-one Mathematics tuition can see that moment in your working and help you fix it. The non-calculator working trainer is useful for practising exact fractions such as 5/13 and 9/12.