Sine Calculator

Enter an angle in degrees or radians to find its sine, cosine, and tangent values instantly.

Quick Facts

Right-triangle definition
sin(θ) = opposite / hypotenuse
Also equals the y-coordinate of the angle's point on the unit circle.
Range
-1 ≤ sin(θ) ≤ 1
Reaches 1 at 90° and -1 at 270°.
Unit conversion
radians = degrees × π/180
A full circle is 360° = 2π radians.

Your Results

Calculated
sin(θ)
-
Sine of the angle
cos(θ)
-
Cosine of the angle
tan(θ)
-
Tangent = sin(θ) / cos(θ)
Angle (other unit)
-
Converted to degrees or radians

Ready

Enter an angle and unit, then press Calculate.

Formula and Method for the Sine of an Angle

For a right triangle, the sine of an angle θ is defined as the ratio of the length of the side opposite that angle to the length of the hypotenuse: sin(θ) = opposite / hypotenuse. This definition extends beyond right triangles through the unit circle: for any angle θ measured counterclockwise from the positive x-axis, sin(θ) is the y-coordinate of the point where the angle's terminal ray crosses a circle of radius 1. This calculator takes an angle in either degrees or radians and returns sin(θ), cos(θ), tan(θ), and the angle converted to the other unit.

How the calculation works

Enter the angle and choose whether it is measured in degrees or radians. Because JavaScript's math functions (and most programming languages) expect radians, the calculator first converts a degree input to radians using radians = degrees × π/180, then evaluates sin(θ) and cos(θ) directly. Tangent is derived from those two results as tan(θ) = sin(θ) / cos(θ). Because cos(θ) = 0 at 90°, 270°, and every angle 90° + 180°n, tangent is undefined at those exact angles and the calculator reports "undefined" rather than a numeric value there.

Degrees vs. radians

Degrees split a full circle into 360 equal parts; radians measure the angle by arc length relative to the radius, so a full circle equals 2π radians (≈6.2832 rad). Common reference points: 30° = π/6 rad, 45° = π/4 rad, 60° = π/3 rad, and 90° = π/2 rad. Always confirm which unit a problem expects before comparing results — the sine of "30" means something very different if it is 30 radians instead of 30 degrees.

Common applications

  • Finding missing sides or angles in right triangles for surveying, construction, and navigation problems
  • Modeling periodic phenomena: sound waves, AC electrical current, tides, and mechanical vibration all follow sinusoidal patterns built from sin(θ)
  • Rotational and oscillatory motion in physics and engineering, including projectile motion components and simple harmonic motion
  • Computer graphics and animation, where sine and cosine generate smooth circular or wave-like movement

Frequently Asked Questions

What is the formula for sine?
In a right triangle, sin(θ) = opposite / hypotenuse. More generally, for any angle θ measured from the positive x-axis on the unit circle, sin(θ) is the y-coordinate of the point where the angle's terminal ray meets the circle.
Does it matter if I enter the angle in degrees or radians?
Yes. Select the correct unit before calculating. Degrees and radians measure the same angle differently (radians = degrees × π/180), and using the wrong unit will produce a completely different sine value.
What is the range of possible sine values?
sin(θ) always falls between -1 and 1 inclusive, since it represents a coordinate on the unit circle. It equals 1 at 90° (π/2 rad), 0 at 0° and 180°, and -1 at 270° (3π/2 rad).
Why is tangent sometimes undefined?
tan(θ) = sin(θ) / cos(θ), so tangent is undefined whenever cos(θ) = 0, which happens at 90°, 270°, and every angle 90° + 180°n.

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