What are Trigonometric Functions?
Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of its sides. They are fundamental to geometry, physics, engineering, and many other sciences. This trigonometric functions calculator instantly computes all six trig functions — sine, cosine, tangent, cosecant, secant, and cotangent — for any angle you enter.
Sine, Cosine, Tangent — The Basic Trig Ratios
In a right triangle, the three primary trig functions are defined in terms of the angle α:
- sin(α) = opposite / hypotenuse — the ratio of the side opposite to the angle to the longest side (hypotenuse).
- cos(α) = adjacent / hypotenuse — the ratio of the side adjacent to the angle to the hypotenuse.
- tan(α) = opposite / adjacent — the ratio of the opposite side to the adjacent side. Also equal to sin(α) / cos(α).
The hypotenuse is always the side opposite the right angle (90°). The adjacent side is the side that forms the angle of interest together with the hypotenuse. The opposite side is across from the angle of interest.
Cosecant, Secant, and Cotangent — The Reciprocal Functions
The other three trig functions are the reciprocals of the basic ones:
- csc(α) = 1 / sin(α) = hypotenuse / opposite
- sec(α) = 1 / cos(α) = hypotenuse / adjacent
- cot(α) = 1 / tan(α) = adjacent / opposite
Note: csc(α) and cot(α) are undefined when sin(α) = 0 (at 0°, 180°, 360°…), and sec(α) and tan(α) are undefined when cos(α) = 0 (at 90°, 270°…). The calculator shows ∞ for these cases.
SOHCAHTOA — The Mnemonic
SOHCAHTOA is a popular memory aid for the three primary trig ratios:
- SOH — Sine = Opposite / Hypotenuse
- CAH — Cosine = Adjacent / Hypotenuse
- TOA — Tangent = Opposite / Adjacent
Measurement Systems — Degrees vs Radians vs Gradians
Angles can be expressed in three different units:
- Degrees (°) — the most common unit in everyday use and in the American measurement system. A full circle = 360°. Right angle = 90°.
- Radians (rad) — the standard unit in mathematics and science (metric / SI system). A full circle = 2π rad ≈ 6.2832 rad. Right angle = π/2 ≈ 1.5708 rad.
- Gradians (grad) — also called grads or gon. A full circle = 400 grad. Right angle = 100 grad. Used in surveying and some European countries.
To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.
Charts of sin, cos, tan, csc, sec, cot
Here are the exact values of all six trig functions at the most common angles:
| Angle (°) | Angle (rad) | sin | cos | tan | csc | sec | cot |
|---|---|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | ∞ | 1 | ∞ |
| 30° | π/6 | 1/2 | √3/2 | 1/√3 | 2 | 2/√3 | √3 |
| 45° | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 | 2/√3 | 2 | 1/√3 |
| 90° | π/2 | 1 | 0 | ∞ | 1 | ∞ | 0 |
| 120° | 2π/3 | √3/2 | −1/2 | −√3 | 2/√3 | −2 | −1/√3 |
| 135° | 3π/4 | √2/2 | −√2/2 | −1 | √2 | −√2 | −1 |
| 150° | 5π/6 | 1/2 | −√3/2 | −1/√3 | 2 | −2/√3 | −√3 |
| 180° | π | 0 | −1 | 0 | ∞ | −1 | ∞ |
| 270° | 3π/2 | −1 | 0 | ∞ | −1 | ∞ | 0 |
| 360° | 2π | 0 | 1 | 0 | ∞ | 1 | ∞ |
How to Use This Trigonometric Functions Calculator
- Select the angle unit — choose Degrees (common in everyday use and US system), Radians (used in math and science), or Gradians.
- Enter the angle value — type any angle, or click one of the quick preset buttons (0°, 30°, 45°, 60°, 90°, etc.).
- Click Calculate — the calculator instantly shows sin, cos, tan, csc, sec, and cot values with their ratios.
Example: For α = 30°:
- sin(30°) = 0.5
- cos(30°) ≈ 0.866
- tan(30°) ≈ 0.577
- csc(30°) = 2
- sec(30°) ≈ 1.155
- cot(30°) ≈ 1.732
What Next?
After finding your trig function values, you might want to explore related calculators:
- Use the Trigonometry Calculator to solve full right triangle problems (find missing sides and angles).
- Use the Law of Sines or Law of Cosines calculators to solve non-right triangles.
- Use the Unit Circle Calculator to visualize angles on the unit circle.
- Use the Arctan Calculator to find an angle from a tangent value (inverse trig).