Sine calculator

What sine means

In a right triangle, the sine of an angle is the side opposite it divided by the hypotenuse:

sin θ = opposite ÷ hypotenuse

That definition only covers angles between 0° and 90°. The unit circle extends it to every angle: turn θ anticlockwise from the positive x-axis, and the point where you land on a circle of radius 1 has coordinates (cos θ, sin θ). Sine is the height of that point, which is why it is positive above the x-axis, negative below it, and never outside −1 to 1.

A worked example

The calculator starts with θ = 30°.

1. 30° is in quadrant I, so it is its own reference angle and the sine is positive.
2. 30° is a standard angle, so its sine is known exactly: sin 30° = 1/2 = 0.5.
3. The point on the unit circle is (√3/2, 1/2) ≈ (0.866, 0.5).

For an angle outside the first quadrant, find the reference angle and then the sign. Take 210°: it is in quadrant III, 30° past 180°, so the reference angle is 30°. Sine is negative in quadrant III, so sin 210° = −sin 30° = −1/2.

Exact values of sine

Every other multiple of 15° reuses one of these values with a sign from its quadrant, and the calculator shows it exactly. Other angles, such as sin 35° = 0.573576, have no simple closed form and are given as decimals.

Inverse sine and the second solution

arcsin undoes sine, but only returns one angle, between −90° and 90°. The equation sin θ = k has two solutions in each full turn (unless k is ±1):

θ = arcsin k + 360°·n or θ = 180° − arcsin k + 360°·n

For sin θ = 0.5 that gives 30° and 150° between 0° and 360°. Forgetting the 180° − α solution is the most common mistake in trigonometric equations. The trig equation solver handles harder cases such as sin 2x = 0.5 or 2sin²x − sin x = 0.

Right triangles: opposite over hypotenuse

In right-triangle mode the default sides are opposite 3 and hypotenuse 5, the 3-4-5 triangle. sin θ = 3 ÷ 5 = 0.6, so θ = arcsin 0.6 = 36.8699° (36°52′12″), and the third side is √(5² − 3²) = 4. A 6 m ladder leaning at 70° reaches 6 × sin 70° = 5.64 m up the wall.

Degrees or radians?

Choose the unit in the calculator, or type it with the number: 30°, 1.2 rad or 50 grad. Anything containing π, such as pi/6 or 5π/4, is read as radians. A bare number in the wrong mode is the usual cause of odd answers: sin 1 is 0.841471 in radians but 0.0174524 in degrees.