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Sine Calculator (sin)

Calculate the sine of any angle in degrees or radians. Shows unit circle coordinates, exact values for special angles, and step-by-step steps.

sin(30°)
0.5
Exact: 1/2
Unit Circle (x, y) = (0.866, 0.5)
Deduction Steps:
Input: sin(30°)
Equivalent: 30° = 0.5236 rad
sin(θ) = y-coordinate on unit circle = 0.5
Exact Value: 1/2
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What Is a Sine Calculator (sin)?

In mathematics, the sine is a fundamental trigonometric function. In a right-angled triangle, the sine of an acute angle is the ratio of the length of the side opposite the angle to the hypotenuse: sin(θ) = Opposite / Hypotenuse.

How to Use This Calculator

  1. Enter the angle θ and select angle mode (Degrees or Radians).
  2. View the decimal sine value and exact special angle representation.
  3. Examine the unit circle coordinates (cos θ, sin θ) and step-by-step derivation.

Sine Trigonometric Definition

\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = y_{\text{unit circle}}

On the unit circle (radius 1), sine corresponds directly to the vertical y-coordinate of the terminal angle point.

Worked Example

Scenario: Calculate sin(30°)

Given: θ = 30° = π / 6 radians

In a 30°-60°-90° triangle, the side opposite 30° is half the hypotenuse.

sin(30°) = 1 / 2 = 0.5

Result: sin(30°) = 0.5 (Exact: 1/2)

Tips & Key Notes

  • The range of the real sine function is strictly bounded between -1 and 1: -1 ≤ sin(θ) ≤ 1.
  • Sine is an odd function: sin(-θ) = -sin(θ).
  • The period of the sine function is 360° (or 2π radians).

Frequently Asked Questions

What are the exact values of sine for common angles?

sin(0°) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1.

What is the period of the sine function?

The sine wave repeats its cycle every 360° or 2π radians: sin(θ + 2π) = sin(θ).

How do you convert degrees to radians?

Multiply degrees by π / 180: radians = degrees × (π / 180).

What is the reciprocal of the sine function?

The reciprocal of sine is cosecant: csc(θ) = 1 / sin(θ).

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