Cosine - Adjacent Over Hypotenuse
Cosine is the second of the three primary trigonometric ratios. Like sine, it connects an angle in a right triangle to the ratio of two of its sides – but this time to the adjacent side rather than the opposite.
The name “cosine” is much younger than “sine” and much less mysterious: the English mathematician Edmund Gunter coined it around 1620 as an abbreviation of complementi sinus, Latin for “sine of the complement”, since cos(θ) always equals sin(90°−θ). The cosine rule itself is far older, first proved in its geometric form by Euclid around 300 BCE, and later generalised algebraically by the 15th-century Persian mathematician Jamshid al-Kashi – which is why in French textbooks it is still called théorème d'Al-Kashi rather than “the cosine rule”.
Definition
For an acute angle θ in a right triangle:
cos(θ) = Adjacent ÷ Hypotenuse
Memory aid: CAH – Cosine = Adjacent over Hypotenuse (the middle part of SOH CAH TOA).
Key Values of Cosine
| Angle (θ) | cos(θ) | Exact value |
|---|---|---|
| 0° | 1 | 1 |
| 30° | ≈ 0.866 | √3 / 2 |
| 45° | ≈ 0.707 | √2 / 2 |
| 60° | 0.5 | 1/2 |
| 90° | 0 | 0 |
Notice that cos(θ) = sin(90° − θ). Cosine and sine are complementary functions – that is exactly where the “co-” prefix comes from.
Using Cosine to Find Missing Sides and Angles
Adjacent = Hypotenuse × cos(θ)
Hypotenuse = Adjacent ÷ cos(θ)
θ = cos−¹(Adjacent ÷ Hypotenuse)
Worked Examples
Adjacent = 12 × cos(50°) = 12 × 0.6428 = 7.71 cm.
cos(θ) = 9/15 = 0.6. θ = cos−¹(0.6) ≈ 53.1°.
Adjacent (horizontal distance) = 10 × cos(60°) = 10 × 0.5 = 5 m.
The Cosine Rule (for any triangle)
For any triangle with sides a, b, c opposite to angles A, B, C:
a² = b² + c² − 2bc·cos(A)
Use the cosine rule when you know two sides and the included angle (SAS) or all three sides (SSS).
Worked Example – Cosine Rule
a² = 8² + 6² − 2(8)(6)cos(60°) = 64 + 36 − 96(0.5) = 100 − 48 = 52.
a = √52 ≈ 7.21 cm.
Key Takeaways
- cos(θ) = Adjacent / Hypotenuse (CAH).
- Key values: cos(0°)=1, cos(30°)=√3/2, cos(45°)=√2/2, cos(60°)=1/2, cos(90°)=0.
- cos(θ) = sin(90°−θ) – sine and cosine are complementary.
- Cosine rule: a² = b² + c² − 2bc·cos(A) for any triangle.