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Year 10 Mathematics · Right-angled trigonometry

Name: ______________________

Date: ____________

1. Which ratio is equal to sin θ?
- ☐ (a) adjacent ÷ hypotenuse
- ☐ (b) opposite ÷ adjacent
- ☐ (c) hypotenuse ÷ opposite
- ☐ (d) opposite ÷ hypotenuse
2. You know the side opposite an angle and the side adjacent to it. Which ratio links them?
- ☐ (a) Sine
- ☐ (b) Cosine
- ☐ (c) Tangent
- ☐ (d) None of these
3. A right-angled triangle has a hypotenuse of 20 cm and an angle of 30°. How long is the side opposite the 30° angle?
- ☐ (a) 10 cm
- ☐ (b) 17.3 cm
- ☐ (c) 34.6 cm
- ☐ (d) 40 cm
4. What is the exact value of tan 45°?
- ☐ (a) 0
- ☐ (b) 1
- ☐ (c) 0.5
- ☐ (d) sqrt(2)
5. A ramp rises 1 m over a horizontal distance of 4 m. What angle does the ramp make with the ground, correct to one decimal place?
- ☐ (a) 75.5°
- ☐ (b) 14.5°
- ☐ (c) 76.0°
- ☐ (d) 14.0°
6. What is the exact value of cos 60°? Give your answer as a decimal or fraction.
7. A right-angled triangle has a hypotenuse of 15 cm and an angle of 50°. Find the side adjacent to the 50° angle in cm, correct to one decimal place.
8. In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 14 cm. Find θ in degrees.

Answer sheet

1. opposite ÷ hypotenuse — SOH: sine is opposite over hypotenuse.
2. Tangent — TOA: tangent is opposite over adjacent.
3. 10 cm — opposite = 20 × sin 30° = 20 × 0.5 = 10 cm. (17.3 cm is the adjacent side, from 20 × cos 30°.)
4. 1 — In a right-angled triangle with a 45° angle, the opposite and adjacent sides are equal, so their ratio is 1.
5. 14.0° — tan θ = 1 ÷ 4 = 0.25, so θ = tan^−1(0.25) ≈ 14.04°, which is 14.0° to one decimal place.
6. 0.5 — cos 60° = 0.5. You can confirm this on a calculator in degree mode.
7. 9.6 cm — adjacent = 15 × cos 50° ≈ 15 × 0.6428 ≈ 9.64, which is 9.6 cm to one decimal place.
8. 30° — sin θ = 7 ÷ 14 = 0.5, so θ = sin^−1(0.5) = 30°.