Trigonometry

In a right-angled triangle, trigonometric ratios relate an angle to the lengths of the triangle's sides. Remembered by SOH-CAH-TOA: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. Given one angle and one side, you can find the other sides — useful for real problems like finding the height of a building from its shadow.

Key terms

Hypotenuse:

The longest side of a right-angled triangle, opposite the right angle.

Opposite / Adjacent:

The side across from the angle (opposite) and the side next to it (adjacent).

SOH-CAH-TOA:

A memory aid for sine, cosine and tangent ratios.

Questions

1. 1. SOH-CAH-TOA helps you remember:

   • Sine, cosine and tangent ratios
   • How to add fractions
   • The order of operations
   • Types of angles

2. 2. The hypotenuse is:

   • The longest side, opposite the right angle
   • The shortest side always
   • A type of angle
   • Always vertical

3. 3. Sine is defined as:

   • Opposite ÷ Hypotenuse
   • Adjacent ÷ Hypotenuse
   • Opposite ÷ Adjacent
   • Hypotenuse ÷ Opposite

4. 4. Cosine is defined as:

   • Adjacent ÷ Hypotenuse
   • Opposite ÷ Hypotenuse
   • Opposite ÷ Adjacent
   • Hypotenuse ÷ Adjacent

5. 5. Tangent is defined as:

   • Opposite ÷ Adjacent
   • Adjacent ÷ Hypotenuse
   • Opposite ÷ Hypotenuse
   • Hypotenuse ÷ Opposite

6. 6. Trigonometry is mainly used with:

   • Right-angled triangles
   • Circles only
   • Squares only
   • Straight lines with no angles

7. 7. The "opposite" side is the one:

   • Across from the angle being used
   • Always the longest side
   • Always the hypotenuse
   • Next to the right angle only

8. 8. A right-angled triangle has a hypotenuse of 10 and an angle of 30°. Which ratio would you use to find the opposite side?

   • Sine
   • Cosine
   • Tangent only
   • None of these

9. 9. A ladder leans against a wall at 60° to the ground, with its foot 2m from the wall. Which ratio relates the angle, the 2m, and the ladder's length?

   • Cosine
   • Sine only
   • None of these
   • Addition only

10. 10. If tan(θ) = opposite/adjacent, and adjacent = 5, opposite = 5, then θ is:

    • 45°
    • 30°
    • 60°
    • 90°

11. 11. A tree casts a shadow of 15m when the sun's angle is 40°. Which ratio would find the tree's height?

    • Tangent
    • Sine only
    • Cosine only
    • None of these

12. 12. In a right triangle with hypotenuse 13 and adjacent side 12, which ratio would find the angle?

    • Cosine
    • Sine only
    • Tangent only
    • None of these

13. 13. Given an angle of 30° and a hypotenuse of 8, the opposite side = 8 × sin(30°) = 8 × 0.5 =:

    • 4
    • 5
    • 6
    • 8

14. 14. The three trigonometric ratios (sine, cosine, tangent) all require:

    • A right-angled triangle
    • A circle
    • Two right angles
    • No angles at all

15. 15. You stand 20m from a tree and measure the angle to its top as 30°. Using tan(30°) ≈ 0.577, the tree's approximate height is:

    • 11.5m
    • 17.3m
    • 20m
    • 34.6m

16. 16. A ramp rises 1.5m over a horizontal distance of 6m. The angle of the ramp (using tan⁻¹) is closest to:

    • 14°
    • 25°
    • 45°
    • 76°

17. 17. A right triangle has an angle of 40° and an adjacent side of 10cm. Using cos(40°) ≈ 0.766, the hypotenuse is approximately:

    • 13.1cm
    • 7.7cm
    • 10cm
    • 15.6cm

18. 18. Why is trigonometry useful for measuring the height of very tall objects, like buildings or mountains?

    • It lets you calculate height from a distance and angle, without direct measurement
    • It requires climbing the object directly
    • It only works on flat objects
    • It has no practical real-world use

19. 19. A kite string is 50m long and makes a 35° angle with the ground. Using sin(35°) ≈ 0.574, the kite's approximate height is:

    • 28.7m
    • 35m
    • 40.9m
    • 50m

20. 20. In a right triangle, if you know two sides but not the angle, which trig-related tool would you use to find the angle?

    • An inverse trig function (sin⁻¹, cos⁻¹ or tan⁻¹)
    • Only addition
    • A protractor and nothing else
    • It cannot be found

21. 21. A surveyor measures a 25° angle to the top of a hill from 100m away on flat ground. Using tan(25°) ≈ 0.466, the hill's approximate height is:

    • 46.6m
    • 25m
    • 100m
    • 4.66m