The Six Trigonometric Functions
Trigonometry studies the relationship between angles and ratios of sides in triangles. There are six trigonometric functions, each relating an angle to a specific ratio. Three are primary (sin, cos, tan) and three are their reciprocals (csc, sec, cot).
Primary: sin(θ) = opp/hyp | cos(θ) = adj/hyp | tan(θ) = opp/adj
Reciprocals: csc(θ) = 1/sin | sec(θ) = 1/cos | cot(θ) = 1/tan
Reciprocals: csc(θ) = 1/sin | sec(θ) = 1/cos | cot(θ) = 1/tan
Complete Exact Values — All Key Angles
| θ | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|
| 0° | 0 | 1 | 0 | undef. | 1 | undef. |
| 30° | 0.5 | 0.866 | 0.577 | 2 | 1.155 | 1.732 |
| 45° | 0.707 | 0.707 | 1 | 1.414 | 1.414 | 1 |
| 60° | 0.866 | 0.5 | 1.732 | 1.155 | 2 | 0.577 |
| 90° | 1 | 0 | undef. | 1 | undef. | 0 |
Key Trigonometric Identities
Pythagorean: sin²θ + cos²θ = 1
Also: 1 + tan²θ = sec²θ
Also: 1 + cot²θ = csc²θ
Double angle: sin(2θ) = 2·sin(θ)·cos(θ)
Double angle: cos(2θ) = cos²θ − sin²θ
Also: 1 + tan²θ = sec²θ
Also: 1 + cot²θ = csc²θ
Double angle: sin(2θ) = 2·sin(θ)·cos(θ)
Double angle: cos(2θ) = cos²θ − sin²θ
The ASTC Rule (Signs by Quadrant)
The ASTC rule (All Students Take Calculus) tells you which functions are positive in each quadrant:
Quadrant I (0°–90°)
All functions positive
Quadrant II (90°–180°)
Sin (and csc) positive
Quadrant III (180°–270°)
Tan (and cot) positive
Quadrant IV (270°–360°)
Cos (and sec) positive
Real-World Applications
- Physics: Resolving forces, projectile motion (sin/cos of launch angle)
- Surveying: Measuring heights of buildings, mountains, and towers using angles
- Engineering: AC circuit analysis uses sin/cos for alternating current
- Computer graphics: Rotation transformations use trig matrices
- Music: Audio waveforms are sums of sine and cosine functions (Fourier analysis)