Sine Calculator

Calculate sine values for any angle in degrees or radians. Visualize sine waves and understand trigonometric properties with interactive graphs.

Sine Function Definition: In a right triangle, sin(θ) = opposite / hypotenuse

On the unit circle, sin(θ) = y-coordinate of the point at angle θ from the positive x-axis

Enter an angle value in your chosen unit
Common angles:
0°
30°
45°
60°
90°
180°
270°
360°
Radians:
0
π
π/2
π/3
π/4
2π
Sine Wave Parameters
Height of the wave (default: 1)
Number of cycles per unit (default: 1)
Horizontal shift (default: 0)
Calculating...

Understanding the Sine Function

The sine function is one of the fundamental trigonometric functions, relating an angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse.

Mathematical Definition:

In a right triangle with angle θ:

sin(θ) = opposite / hypotenuse

On the unit circle (radius = 1):

sin(θ) = y-coordinate

The sine function is periodic with period 2π radians (360°), and its values range from -1 to 1.

Sine Function Properties

Property Description Mathematical Expression
Domain All real numbers (-∞, ∞)
Range Values between -1 and 1 [-1, 1]
Period Repeats every 2π radians sin(θ + 2π) = sin(θ)
Amplitude Maximum deviation from center |A| in A·sin(x)
Symmetry Odd function sin(-θ) = -sin(θ)
Zeros Where sin(θ) = 0 θ = nπ, n ∈ ℤ
Maximum sin(θ) = 1 θ = π/2 + 2nπ
Minimum sin(θ) = -1 θ = 3π/2 + 2nπ

Common Sine Values

Angle (degrees) Angle (radians) sin(θ) Exact Value
0° 0 0 0
30° π/6 0.5 1/2
45° π/4 0.7071 √2/2
60° π/3 0.8660 √3/2
90° π/2 1 1
180° π 0 0
270° 3π/2 -1 -1
360° 2π 0 0

Sine Function Applications

1

Wave Motion: Sine waves describe oscillatory phenomena like sound waves, light waves, and alternating current.

2

Circular Motion: The vertical position of an object in uniform circular motion follows a sine function over time.

3

Signal Processing: Sine functions are used in Fourier analysis to represent complex signals as sums of simple sine waves.

4

Geometry & Navigation: The sine law relates sides and angles in any triangle, essential for triangulation and GPS calculations.

Frequently Asked Questions

The sine function takes an angle and returns a ratio between -1 and 1. The arcsine (or inverse sine) function takes a ratio between -1 and 1 and returns an angle. They are inverse functions: if y = sin(θ), then θ = arcsin(y).

Sine values represent the y-coordinate on the unit circle, where the radius is 1. Since the maximum vertical distance from the center is 1 (at the top and bottom of the circle), sine values range from -1 to 1.

For common angles (0°, 30°, 45°, 60°, 90°), memorize the exact values. For other angles, you can use trigonometric identities, approximation formulas, or reference angles. Historically, sine tables were used before calculators.

Sine and cosine are cofunctions: sin(θ) = cos(90° - θ) or sin(θ) = cos(π/2 - θ). On the unit circle, cosine represents the x-coordinate while sine represents the y-coordinate. They are phase-shifted versions of each other: cos(θ) = sin(θ + π/2).

For real angles, sine values are always between -1 and 1. However, for complex numbers, sine can exceed this range. In the context of right triangles, the ratio opposite/hypotenuse cannot exceed 1 since the hypotenuse is always the longest side.

Key Trigonometric Identities

Quick Tips

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