Definitions
In a right triangle with angle θ: sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent (the SOH-CAH-TOA mnemonic).
On the unit circle (radius 1, centered at the origin), the point reached by turning an angle θ counterclockwise from the positive x-axis has coordinates (cos θ, sin θ), and tan θ = sin θ ÷ cos θ. This extends the functions to every angle, including negative ones and angles beyond 90°.
The reciprocal functions are csc θ = 1/sin θ, sec θ = 1/cos θ and cot θ = 1/tan θ.
Sources: OpenStax: Precalculus 2e §5.2 Unit Circle: Sine and Cosine Functions; OpenStax: Precalculus 2e §5.3 The Other Trigonometric Functions; NIST Digital Library of Mathematical Functions: DLMF §4.14 Trigonometric Functions: Definitions and Periodicity
Exact values from 0° to 360°
Exact values come from the 30-60-90 and 45-45-90 triangles and the angle-sum formulas (for 15° and 75°); the sign in each quadrant follows the unit circle (“All Students Take Calculus”: all positive in quadrant I, sine in II, tangent in III, cosine in IV). Decimals are to six places.
| Degrees | Radians | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 15° | π/12 | (√6 − √2)/4 ≈ 0.258819 | (√6 + √2)/4 ≈ 0.965926 | 2 − √3 ≈ 0.267949 |
| 30° | π/6 | 1/2 ≈ 0.500000 | √3/2 ≈ 0.866025 | √3/3 ≈ 0.577350 |
| 45° | π/4 | √2/2 ≈ 0.707107 | √2/2 ≈ 0.707107 | 1 |
| 60° | π/3 | √3/2 ≈ 0.866025 | 1/2 ≈ 0.500000 | √3 ≈ 1.732051 |
| 75° | 5π/12 | (√6 + √2)/4 ≈ 0.965926 | (√6 − √2)/4 ≈ 0.258819 | 2 + √3 ≈ 3.732051 |
| 90° | π/2 | 1 | 0 | Undefined |
| 105° | 7π/12 | (√6 + √2)/4 ≈ 0.965926 | −(√6 − √2)/4 ≈ −0.258819 | −(2 + √3) ≈ −3.732051 |
| 120° | 2π/3 | √3/2 ≈ 0.866025 | −1/2 ≈ −0.500000 | −√3 ≈ −1.732051 |
| 135° | 3π/4 | √2/2 ≈ 0.707107 | −√2/2 ≈ −0.707107 | −1 |
| 150° | 5π/6 | 1/2 ≈ 0.500000 | −√3/2 ≈ −0.866025 | −√3/3 ≈ −0.577350 |
| 165° | 11π/12 | (√6 − √2)/4 ≈ 0.258819 | −(√6 + √2)/4 ≈ −0.965926 | −(2 − √3) ≈ −0.267949 |
| 180° | π | 0 | −1 | 0 |
| 195° | 13π/12 | −(√6 − √2)/4 ≈ −0.258819 | −(√6 + √2)/4 ≈ −0.965926 | 2 − √3 ≈ 0.267949 |
| 210° | 7π/6 | −1/2 ≈ −0.500000 | −√3/2 ≈ −0.866025 | √3/3 ≈ 0.577350 |
| 225° | 5π/4 | −√2/2 ≈ −0.707107 | −√2/2 ≈ −0.707107 | 1 |
| 240° | 4π/3 | −√3/2 ≈ −0.866025 | −1/2 ≈ −0.500000 | √3 ≈ 1.732051 |
| 255° | 17π/12 | −(√6 + √2)/4 ≈ −0.965926 | −(√6 − √2)/4 ≈ −0.258819 | 2 + √3 ≈ 3.732051 |
| 270° | 3π/2 | −1 | 0 | Undefined |
| 285° | 19π/12 | −(√6 + √2)/4 ≈ −0.965926 | (√6 − √2)/4 ≈ 0.258819 | −(2 + √3) ≈ −3.732051 |
| 300° | 5π/3 | −√3/2 ≈ −0.866025 | 1/2 ≈ 0.500000 | −√3 ≈ −1.732051 |
| 315° | 7π/4 | −√2/2 ≈ −0.707107 | √2/2 ≈ 0.707107 | −1 |
| 330° | 11π/6 | −1/2 ≈ −0.500000 | √3/2 ≈ 0.866025 | −√3/3 ≈ −0.577350 |
| 345° | 23π/12 | −(√6 − √2)/4 ≈ −0.258819 | (√6 + √2)/4 ≈ 0.965926 | −(2 − √3) ≈ −0.267949 |
| 360° | 2π | 0 | 1 | 0 |
Sources: OpenStax: Precalculus 2e §5.2 Unit Circle: Sine and Cosine Functions
Key identities
- Pythagorean: sin²θ + cos²θ = 1; 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ
- Even and odd: sin(−θ) = −sin θ; cos(−θ) = cos θ; tan(−θ) = −tan θ
- Cofunction: sin(90° − θ) = cos θ; cos(90° − θ) = sin θ
- Sum: 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 = 2 sin θ cos θ; cos 2θ = cos²θ − sin²θ = 1 − 2sin²θ
- Tangent sum: tan(a + b) = (tan a + tan b) ÷ (1 − tan a tan b)
Sources: NIST Digital Library of Mathematical Functions: DLMF §4.21 Trigonometric Functions: Identities
Graphs of sin, cos and tan
y = sin x is a wave with period 2π (360°) and amplitude 1, ranging over [−1, 1]; it crosses zero at every multiple of π and peaks at π/2 + 2πk.
y = cos x is the same wave shifted left by π/2: cos x = sin(x + π/2). It starts at its maximum, 1, at x = 0.
y = tan x has period π (180°), takes every real value, crosses zero at multiples of π and has vertical asymptotes at x = π/2 + πk, where cos x = 0.
Sources: OpenStax: Precalculus 2e §6.1 Graphs of the Sine and Cosine Functions; OpenStax: Calculus Volume 1 §1.3 Trigonometric Functions