Sin(θ) Calculator

Compute sin(θ) with unit circle position, quadrant, and related trig values.

Enter Angle θ

Result

sin(θ)
--
θ in Degrees--
θ in Radians--
sin(θ)--
cos(θ)--
tan(θ)--
Quadrant--
Unit Circle (x, y)--
Reference Angle--
sin²(θ) + cos²(θ)--

Step-by-Step Solution

Understanding Sin(θ)

The sine of an angle θ is one of the primary trigonometric functions. On the unit circle, sin(θ) equals the y-coordinate of the point where the terminal side of the angle intersects the circle. In a right triangle, sin(θ) is the ratio of the opposite side to the hypotenuse.

Common Sine Values

sin(0°) = 0

At 0 degrees, the point on the unit circle is (1, 0).

sin(0) = 0

sin(30°) = 1/2

At 30 degrees (π/6), the sine value is exactly 0.5.

sin(π/6) = 1/2

sin(45°) = √2/2

At 45 degrees (π/4), sine equals the square root of 2 over 2.

sin(π/4) = √2/2

sin(60°) = √3/2

At 60 degrees (π/3), sine equals the square root of 3 over 2.

sin(π/3) = √3/2

sin(90°) = 1

At 90 degrees (π/2), sine reaches its maximum value of 1.

sin(π/2) = 1

sin(180°) = 0

At 180 degrees (π), the sine returns to zero.

sin(π) = 0

The Unit Circle

The unit circle is a circle with radius 1 centered at the origin. For any angle θ, the corresponding point on the unit circle is (cos(θ), sin(θ)). This provides a geometric way to understand trigonometric functions for all angles, not just those in right triangles.

Quadrant Signs

  • Quadrant I (0° to 90°): sin is positive, cos is positive
  • Quadrant II (90° to 180°): sin is positive, cos is negative
  • Quadrant III (180° to 270°): sin is negative, cos is negative
  • Quadrant IV (270° to 360°): sin is negative, cos is positive