Sine Cosine Tangent Calculator

Enter an angle in degrees or radians to get its sine, cosine, and tangent (sin θ, cos θ, tan θ), plus the same angle converted to the other unit.

Quick Facts

Sine
sin θ = opposite / hypotenuse
Also the y-coordinate of the point at angle θ on the unit circle.
Cosine
cos θ = adjacent / hypotenuse
Also the x-coordinate of the point at angle θ on the unit circle.
Tangent
tan θ = sin θ / cos θ
Undefined whenever cos θ = 0, e.g. at 90° and 270°.
Pythagorean identity
sin²θ + cos²θ = 1
True for every angle θ; the basis for most trig simplifications.

Your Results

Calculated
Sine (sin θ)
-
y-coordinate on the unit circle
Cosine (cos θ)
-
x-coordinate on the unit circle
Tangent (tan θ)
-
sin θ / cos θ
Angle in other unit
-
Degrees ↔ radians conversion

Ready

Enter an angle and unit, then press Calculate.

Formula and Method for Sine, Cosine, and Tangent

Sine, cosine, and tangent are the three basic trigonometric functions that relate an angle to the ratios of a right triangle's sides — and, more generally, to the coordinates of a point on the unit circle. For an angle θ: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent = sin θ / cos θ. This calculator takes an angle in either degrees or radians and returns all three ratios, along with the angle converted to the other unit.

The unit circle definition

The right-triangle definitions only work cleanly for angles between 0° and 90°. To handle any angle — negative, obtuse, or larger than a full turn — trigonometry uses the unit circle: a circle of radius 1 centered at the origin. For an angle θ measured counterclockwise from the positive x-axis, the point where the angle's ray crosses the circle has coordinates (cos θ, sin θ). Tangent is then the slope of that ray, tan θ = sin θ / cos θ = y / x. This calculator uses the unit circle definition internally, so it works for any real angle, not just those inside a triangle.

Degrees, radians, and undefined tangent values

Angles can be measured in degrees (a full circle = 360°) or radians (a full circle = 2π ≈ 6.2832 rad). Convert with radians = degrees × π/180 and degrees = radians × 180/π. Tangent is undefined wherever cos θ = 0 — that is, at 90°, 270°, and every 180° step from those points (θ = 90° + n·180°) — because dividing by zero has no defined value; the calculator reports "Undefined" rather than a number at those angles.

Common applications

  • Finding missing sides or angles in right triangles (surveying, construction, navigation)
  • Modeling periodic phenomena: sound waves, AC current, tides, and mechanical vibration all follow sinusoidal patterns based on sine and cosine
  • Rotational and circular motion in physics and engineering, where position, velocity, and force components are resolved using sin θ and cos θ
  • Computer graphics and animation, where sine and cosine generate smooth circular or wave-like motion

Frequently Asked Questions

What is the difference between sine, cosine, and tangent?
For an angle θ in a right triangle, sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, and tan θ = opposite / adjacent = sin θ / cos θ. On the unit circle, cos θ and sin θ are simply the x- and y-coordinates of the point at angle θ.
How do I convert between degrees and radians?
Multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. For example, 90° = 90 × π/180 = π/2 ≈ 1.5708 radians.
Why is tangent undefined at 90° and 270°?
tan θ equals sin θ divided by cos θ, and cos θ = 0 at 90°, 270°, and every 180° interval from those angles. Dividing by zero makes the tangent undefined at those points (a vertical asymptote on its graph).
What does a negative sine or cosine value mean?
The sign tells you which quadrant the angle falls in: sine is positive in quadrants I and II and negative in III and IV; cosine is positive in I and IV and negative in II and III. The mnemonic "All Students Take Calculus" lists which of sin, cos, and tan are positive in each quadrant (All, Sin, Tan, Cos).

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