Sine vs Cosine: What is the Difference?
Sine and cosine are the two fundamental trigonometric functions. Both describe ratios of sides in a right triangle and oscillating waves on the unit circle — but they describe different sides and are shifted relative to each other.
In a right triangle: sin(θ) = Opposite/Hypotenuse; cos(θ) = Adjacent/Hypotenuse. On the unit circle: sin(θ) is the y-coordinate and cos(θ) is the x-coordinate of the point at angle θ. The key relationship: cos(θ) = sin(θ + 90°) — cosine is just sine shifted 90° to the left.
| Property | Sine (sin) | Cosine (cos) |
|---|---|---|
| Right triangle definition | Opposite / Hypotenuse | Adjacent / Hypotenuse |
| Unit circle | y-coordinate | x-coordinate |
| Value at 0° | 0 | 1 |
| Value at 30° | 1/2 = 0.5 | √3/2 ≈ 0.866 |
| Value at 45° | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 |
| Value at 60° | √3/2 ≈ 0.866 | 1/2 = 0.5 |
| Value at 90° | 1 | 0 |
| Value at 180° | 0 | −1 |
| Period | 360° (2π) | 360° (2π) |
| Range | [−1, 1] | [−1, 1] |
| Even or odd? | Odd: sin(−θ) = −sin(θ) | Even: cos(−θ) = cos(θ) |
| Graph starts at | (0, 0) — zero at origin | (0, 1) — max at origin |
| Phase relationship | sin(θ) = cos(90° − θ) | cos(θ) = sin(90° − θ) |
| Derivative | d/dθ sin(θ) = cos(θ) | d/dθ cos(θ) = −sin(θ) |
Definitions in a Right Triangle
sin(θ): The ratio of the side Opposite angle θ to the Hypotenuse. In the SOH of SOH-CAH-TOA: Sin = Opposite/Hypotenuse.
cos(θ): The ratio of the side Adjacent to angle θ to the Hypotenuse. In the CAH of SOH-CAH-TOA: Cos = Adjacent/Hypotenuse.
These two ratios use the hypotenuse (largest side) as the denominator. The key difference is which leg (opposite vs adjacent) goes in the numerator. For the same right triangle, if you label one acute angle θ and the other φ, then sin(θ) = cos(φ) and cos(θ) = sin(φ) — because what's "opposite" to one angle is "adjacent" to the other.
The Unit Circle — Cosine is x, Sine is y
On the unit circle (radius = 1), any point at angle θ from the positive x-axis has coordinates (cos θ, sin θ). This is the most powerful way to understand both functions — it extends them beyond right triangles to all angles, including negative angles and angles greater than 90°.
sin(θ) = y-coordinate: starts at 0 (at θ=0°), rises to maximum 1 (at 90°), returns to 0 (at 180°), falls to −1 (at 270°), and returns to 0 (at 360°).
cos(θ) = x-coordinate: starts at 1 (at θ=0°), falls to 0 (at 90°), falls to −1 (at 180°), returns to 0 (at 270°), and returns to 1 (at 360°). Cosine is sine shifted 90° left.
Key Identities Connecting Sine and Cosine
Pythagorean identity: sin²(θ) + cos²(θ) = 1 for all θ. This follows from the unit circle: x² + y² = 1.
Co-function identity: sin(θ) = cos(90°−θ) and cos(θ) = sin(90°−θ). Sine and cosine are "co-functions" — sin(30°) = cos(60°), sin(45°) = cos(45°).
Phase shift: cos(θ) = sin(θ + 90°). Cosine is sine shifted 90° to the left. Or: sin(θ) = cos(θ − 90°).
Angle addition: sin(A+B) = sin(A)cos(B) + cos(A)sin(B). cos(A+B) = cos(A)cos(B) − sin(A)sin(B).
Double angle: sin(2θ) = 2sin(θ)cos(θ). cos(2θ) = cos²(θ) − sin²(θ) = 2cos²(θ)−1 = 1−2sin²(θ).
When to Use Sine vs Cosine
Use sine when you have the angle and the hypotenuse and want the opposite side: opposite = H × sin(θ). Also use sine when computing the vertical component of a vector at angle θ from horizontal.
Use cosine when you have the angle and the hypotenuse and want the adjacent side: adjacent = H × cos(θ). Also use cosine when computing the horizontal component of a vector at angle θ from horizontal.
Both together: To decompose a force F at angle θ: horizontal component = F·cos(θ), vertical component = F·sin(θ). This is used in physics for projectile motion, inclined planes, and vector resolution.
Verdict
Sine and cosine are essentially the same function shifted by 90°. Both have the same range [−1, 1], the same period (360°), and both satisfy sin²+cos²=1. The key difference is their starting point: cosine starts at its maximum (1) at θ=0, while sine starts at 0.
- ✓sin(θ) = opposite/hypotenuse; cos(θ) = adjacent/hypotenuse in right triangles.
- ✓On the unit circle: sin = y-coordinate, cos = x-coordinate.
- ✓cos(θ) = sin(θ + 90°): cosine is just sine shifted left by 90°.
- ✓Both are odd (sin) / even (cos) functions: sin(−θ) = −sin(θ); cos(−θ) = cos(θ).
- ✓Derivatives cycle: d(sin)/dθ = cos; d(cos)/dθ = −sin; d²(sin)/dθ² = −sin.