Sine, Cosine, and Tangent Explained
Sine (sin), cosine (cos), and tangent (tan) are the three primary trigonometric functions. They describe the relationship between the angles and sides of a right-angled triangle, and are extended to all angles via the unit circle.
In a right triangle with angle θ:
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)
sin(θ) = Opposite / Hypotenuse
cos(θ) = Adjacent / Hypotenuse
tan(θ) = Opposite / Adjacent = sin(θ) / cos(θ)
Memory Trick: SOH-CAH-TOA
The most famous mnemonic in trigonometry:
SOH – Sin = Opposite / Hypotenuse
CAH – Cos = Adjacent / Hypotenuse
TOA – Tan = Opposite / Adjacent
CAH – Cos = Adjacent / Hypotenuse
TOA – Tan = Opposite / Adjacent
Exact Values for Common Angles
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 = 0.5 | √3/2 ≈ 0.866 | 1/√3 ≈ 0.577 |
| 45° | π/4 | √2/2 ≈ 0.707 | √2/2 ≈ 0.707 | 1 |
| 60° | π/3 | √3/2 ≈ 0.866 | 1/2 = 0.5 | √3 ≈ 1.732 |
| 90° | π/2 | 1 | 0 | undefined |
| 120° | 2π/3 | √3/2 ≈ 0.866 | -0.5 | -√3 ≈ -1.732 |
| 180° | π | 0 | -1 | 0 |
| 270° | 3π/2 | -1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
The Other Three Trig Functions
The three primary functions have reciprocals that are also widely used:
cosecant: csc(θ) = 1/sin(θ)
secant: sec(θ) = 1/cos(θ)
cotangent: cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
secant: sec(θ) = 1/cos(θ)
cotangent: cot(θ) = 1/tan(θ) = cos(θ)/sin(θ)
The Unit Circle
The unit circle (radius = 1) centered at the origin extends trig functions to all angles. For any angle θ, the point on the unit circle is (cos θ, sin θ). This means:
- cos θ = x-coordinate on the unit circle
- sin θ = y-coordinate on the unit circle
- sin²θ + cos²θ = 1 (the Pythagorean identity — always true)
Worked Example – Finding a Triangle Side
A 10 m ladder leans against a wall at 60° to the ground. How high up the wall does it reach?
Height = Hypotenuse × sin(angle) = 10 × sin(60°) = 10 × 0.866 = 8.66 m
Real-World Applications
- Engineering: Force resolution, beam angle analysis, stress calculations
- Architecture: Roof pitch, ramp gradients, staircase angles
- Navigation: Bearing calculations, GPS coordinates, ship/aircraft headings
- Physics: Wave motion, pendulums, oscillations, optics (Snell's Law)
- Computer graphics: Rotation matrices, 3D rendering, game physics
- Music: Sound waves are modeled using sine functions