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.
The word “sine” has an unusually tangled history. Indian astronomers, including Aryabhata around 500 CE, worked with a quantity called jya (meaning “chord”) in half-chord tables used for astronomy. Arabic scholars transliterated jya as jiba, a word with no real meaning in Arabic; later translators, reading it without vowels as jaib (meaning “bay” or “fold of a garment”), rendered it into Latin as sinus – and that Latin word for “bay” is where our modern “sine” comes from. The translation, made around 1145 CE by the English scholar Robert of Chester, is a striking example of how a simple transcription error can lodge itself permanently in mathematical language.
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).