Sine - Opposite Over Hypotenuse
Sine is one of the three primary trigonometric ratios. It connects an angle in a right triangle to the ratio of two of its sides. Once you know the sine of an angle, you can find missing side lengths and angles in any right triangle.
Definition
For an acute angle θ in a right triangle:
sin(θ) = Opposite ÷ Hypotenuse
A useful memory aid: SOH – Sine = Opposite over Hypotenuse (part of SOH CAH TOA).
Key Values of Sine
| Angle (θ) | sin(θ) | Exact value |
|---|---|---|
| 0° | 0 | 0 |
| 30° | 0.5 | 1/2 |
| 45° | ≈ 0.707 | √2 / 2 |
| 60° | ≈ 0.866 | √3 / 2 |
| 90° | 1 | 1 |
Using Sine to Find a Missing Side
If you know an angle and the hypotenuse:
Opposite = Hypotenuse × sin(θ)
If you know an angle and the opposite side:
Hypotenuse = Opposite ÷ sin(θ)
Using Inverse Sine to Find a Missing Angle
If you know the opposite and hypotenuse but not the angle:
θ = sin−¹(Opposite ÷ Hypotenuse)
sin−¹ is also written arcsin. Use the sin−¹ button on your calculator.
Worked Examples
Opposite = 10 × sin(35°) = 10 × 0.5736 = 5.74 cm.
sin(θ) = 7 / 14 = 0.5. θ = sin−¹(0.5) = 30°.
Hypotenuse = 9 ÷ sin(45°) = 9 ÷ (√2/2) = 9 × (2/√2) = 9√2 ≈ 12.73 cm.
sin(θ) = 3/6 = 0.5. θ = sin−¹(0.5) = 30°.
The Sine Rule (for any triangle)
For any triangle with sides a, b, c opposite to angles A, B, C:
a / sin(A) = b / sin(B) = c / sin(C)
Use the sine rule when you know two angles and one side, or two sides and a non-included angle.
Key Takeaways
- sin(θ) = Opposite / Hypotenuse (SOH).
- Key values: sin(30°)=1/2, sin(45°)=√2/2, sin(60°)=√3/2, sin(90°)=1.
- To find a missing angle use θ = sin−¹(ratio).
- The sine rule extends sine to any triangle: a/sin(A) = b/sin(B) = c/sin(C).
Practice Questions
- Find the opposite side: hypotenuse = 15 cm, θ = 40°.
- Find the hypotenuse: opposite = 8 cm, θ = 30°.
- Find angle θ: opposite = 5 cm, hypotenuse = 13 cm.
- A kite string is 50 m long and makes a 60° angle with the ground. How high is the kite?
- In triangle ABC: A = 40°, B = 75°, a = 10 cm. Use the sine rule to find b.