Angle Input
All Six Trig Functions
| Function | Exact / Fraction | Decimal (8 dp) |
|---|
Unit Circle Diagram
Inverse Trigonometric Functions
Select Known Values
Given all three sides a, b, c — find all angles and area.
Convert Angle
Identity Type
The Six Trigonometric Functions
Trigonometry defines six functions relating the angles and sides of triangles. For a right triangle with angle θ, hypotenuse H, opposite side O, and adjacent side A:
- sin(θ) = O/H — the sine function
- cos(θ) = A/H — the cosine function
- tan(θ) = O/A = sin/cos — the tangent function
- csc(θ) = H/O = 1/sin — cosecant (reciprocal of sine)
- sec(θ) = H/A = 1/cos — secant (reciprocal of cosine)
- cot(θ) = A/O = 1/tan — cotangent (reciprocal of tangent)
Special Angle Values
These exact values appear frequently and should be memorized:
| θ (deg) | θ (rad) | sin θ | cos θ | tan θ |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| 180° | π | 0 | −1 | 0 |
| 270° | 3π/2 | −1 | 0 | undefined |
| 360° | 2π | 0 | 1 | 0 |
Law of Sines and Law of Cosines
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Used when you know AAS, ASA, or SSA (ambiguous case).
Law of Cosines: c² = a² + b² − 2ab·cos(C). Generalises the Pythagorean theorem and is used for SSS and SAS cases. Rearranging: cos(A) = (b² + c² − a²) / (2bc).
Heron's Formula: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter.
Frequently Asked Questions
SOHCAHTOA is a mnemonic for the three basic trig ratios in a right triangle: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It only applies to right-angled triangles.
The three reciprocal functions are cosecant (csc = 1/sin), secant (sec = 1/cos), and cotangent (cot = 1/tan). They are the multiplicative inverses of the primary functions and are useful in integration, optics, and engineering.
The law of cosines states c² = a² + b² − 2ab·cos(C), where C is the angle opposite side c. It generalises the Pythagorean theorem (which is the special case C = 90°) to any triangle. Use it when you know SSS or SAS.
A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. There are 2π radians in a full circle, so 1 rad ≈ 57.2958°. Radians are the preferred unit in calculus because they make derivative formulas simpler (d/dx sin x = cos x only holds when x is in radians).