What are trigonometric functions?

Trigonometric functions (sin, cos, tan) relate angles of a right triangle to the ratios of its sides. Sine (sin) = opposite/hypotenuse, Cosine (cos) = adjacent/hypotenuse, Tangent (tan) = opposite/adjacent.

What is the difference between degrees and radians?

Degrees and radians are two ways to measure angles. A full circle is 360 degrees or 2π radians. To convert: radians = degrees × π/180, degrees = radians × 180/π.

What are the values of sin, cos, and tan for common angles?

For 0°: sin=0, cos=1, tan=0. For 30°: sin=0.5, cos=√3/2≈0.866, tan=1/√3≈0.577. For 45°: sin=cos=√2/2≈0.707, tan=1. For 90°: sin=1, cos=0, tan=undefined.

Why is tangent undefined for 90 degrees?

Tangent is undefined at 90° (and 270°) because tan = sin/cos, and cos(90°) = 0. Division by zero is undefined in mathematics.

What are inverse trigonometric functions?

Inverse trig functions (arcsin, arccos, arctan) find the angle when you know the ratio. For example, if sin(θ) = 0.5, then arcsin(0.5) = 30° (or π/6 radians).

Where are trigonometric functions used?

Trigonometry is used in physics (waves, oscillations), engineering (signal processing, construction), navigation (GPS, compass), computer graphics, and many fields involving angles and periodic phenomena.

Trigonometry Calculator

Calculate sine, cosine, and tangent for any angle. Support for both degrees and radians with accurate trigonometric function values.

Calculator

Result

Sine (sin): -
Cosine (cos): -
Tangent (tan): -

Formula & Guide

Formulas

sin

Sine

sin(θ) = opposite / hypotenuse

Ratio of opposite side to hypotenuse

cos

Cosine

cos(θ) = adjacent / hypotenuse

Ratio of adjacent side to hypotenuse

tan

Tangent

tan(θ) = opposite / adjacent

Ratio of opposite to adjacent side

=

Unit Conversion

radians = degrees × π/180

Convert degrees to radians

Formula Variables

θ

Angle

The angle in degrees or radians

opposite

Opposite Side

The side opposite to the angle in a right triangle

adjacent

Adjacent Side

The side adjacent to the angle (not the hypotenuse)

hypotenuse

Hypotenuse

The longest side of a right triangle, opposite the right angle

Step-by-Step Scenario

Example Scenario

Angle

30°

1

Convert to Radians (if needed)

30° × π/180 = π/6 radians ≈ 0.524 radians

Many calculators work directly with degrees, but internally convert to radians

2

Calculate Trigonometric Functions

sin(30°) = 0.5
cos(30°) = √3/2 ≈ 0.866

tan(30°) = 1/√3 ≈ 0.577

Additional Examples

45 Degrees

Angle: 45°

sin(45°)

√2/2 ≈ 0.707

cos(45°)

√2/2 ≈ 0.707

tan(45°)

1

60 Degrees

Angle: 60°

sin(60°)

√3/2 ≈ 0.866

cos(60°)

0.5

tan(60°)

√3 ≈ 1.732

Characteristics of Trigonometry

Periodic Functions

Trigonometric functions are periodic, repeating their values every 360° (2π radians). This makes them ideal for modeling cyclical phenomena.

Right Triangle Relationships

Trig functions relate angles to side ratios in right triangles, making them fundamental to geometry and measurement.

Wide Applications

Used in physics (waves, oscillations), engineering (signal processing), navigation, computer graphics, and many scientific fields.

Unit Flexibility

Can work with degrees (0-360) or radians (0-2π). Radians are preferred in advanced mathematics and physics.

Important Notes

Trigonometric functions are periodic: sin(θ + 360°) = sin(θ), cos(θ + 360°) = cos(θ), and tan(θ + 180°) = tan(θ).
Tangent is undefined at 90° and 270° (and their radian equivalents) because cos equals zero, causing division by zero.
The range of sin and cos is [-1, 1], while tan can be any real number (except where undefined).
To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.
Common angle values: 0°, 30°, 45°, 60°, 90° have exact trigonometric values that are important to memorize.

Frequently Asked Questions

Find answers to common questions about trigonometric calculations.