Sin Theta Calculator

Enter an angle to compute sin(θ), along with cos(θ), tan(θ), and the angle's quadrant using the unit-circle definition of sine.

Quick Facts

Right-triangle definition
sin(θ) = opposite / hypotenuse
Valid for acute angles inside a right triangle.
Unit-circle definition
sin(θ) = y-coordinate
The y-coordinate of the point at angle θ on a circle of radius 1.
Range
-1 ≤ sin(θ) ≤ 1
Sine never exceeds 1 or drops below -1.
Periodicity
sin(θ + 360°) = sin(θ)
Sine repeats every full revolution (2π radians).

Your Results

Calculated
sin(θ)
-
y-coordinate on the unit circle
cos(θ)
-
x-coordinate on the unit circle
tan(θ)
-
sin(θ) / cos(θ)
Quadrant & reference angle
-
Location of θ on the unit circle

Ready

Enter an angle and unit, then press Calculate.

Formula and Method for Sin Theta

Sine is one of the three primary trigonometric functions. In a right triangle, sin(θ) = opposite ÷ hypotenuse — the length of the side opposite angle θ divided by the length of the hypotenuse. This definition only works for angles between 0° and 90°, so a more general definition uses the unit circle: for any angle θ measured counter-clockwise from the positive x-axis, sin(θ) is the y-coordinate of the point where the angle's terminal side crosses a circle of radius 1. This calculator uses the unit-circle definition, so it works for any angle, positive or negative, in degrees, radians, or gradians.

How the calculation works

Enter the angle θ and pick its unit. The calculator first converts the angle to radians internally (radians = degrees × π/180, or radians = gradians × π/200), then evaluates sin(θ), cos(θ), and tan(θ) = sin(θ)/cos(θ) using those standard trigonometric functions. It also normalizes the angle to a value between 0° and 360° to report which quadrant the angle falls in and the corresponding reference angle — the acute angle between the terminal side and the x-axis, which shares the same sine magnitude as θ.

Common mistakes

  • Mixing degrees and radians: sin(30) means something very different depending on whether 30 is degrees (sin 30° = 0.5) or radians (sin 30 rad ≈ -0.988). Always confirm the unit before reading a result.
  • Confusing the reference angle with θ itself: the reference angle tells you the sine's magnitude, but the sign still depends on the quadrant — sine is positive in quadrants I and II, negative in quadrants III and IV.
  • Assuming tan(θ) is always defined: because tan(θ) = sin(θ)/cos(θ), it is undefined at 90°, 270°, and every 180° increment from those angles, where cos(θ) = 0.

Common applications

  • Finding missing sides or angles in right triangles for surveying, construction, and navigation
  • Modeling periodic phenomena: sound waves, AC electrical current, tides, and mechanical vibration all follow sinusoidal (sine-based) patterns
  • Rotational and projectile motion in physics, where sin(θ) resolves a vector into its vertical component
  • Computer graphics and animation, where sine functions drive smooth oscillating motion

Frequently Asked Questions

What is the formula for sin theta?
In a right triangle, sin(θ) = opposite ÷ hypotenuse. More generally, on the unit circle, sin(θ) is the y-coordinate of the point where the terminal side of angle θ meets the circle.
How do I convert degrees to radians for a sine calculation?
Multiply the degree value by π/180 (about 0.017453). For example, 30° × π/180 ≈ 0.5236 radians, and sin(30°) = sin(0.5236 rad) = 0.5.
What is the range of sin(θ)?
Sin(θ) always falls between -1 and 1 inclusive, because it equals a coordinate on the unit circle, which has radius 1.
Why is tan(θ) sometimes undefined?
Because tan(θ) = sin(θ) / cos(θ), it is undefined whenever cos(θ) = 0 — at 90°, 270°, and every 180° increment from those points (odd multiples of 90°).

Related calculators