Sine Calculator | Calculate Sine of an Angle (Sin(x))

What is the Sine Function?

Trigonometric Foundation
The Unit Circle
Properties of the Sine Wave

The sine function, denoted as sin(x), is one of the primary trigonometric functions. It is fundamentally the ratio of the length of the side of a right-angled triangle that is opposite the angle to the length of the hypotenuse. The value of sine is always between -1 and 1.

Visualizing Sine on the Unit Circle

The unit circle (a circle with a radius of 1) provides a powerful way to visualize the sine function. For any point (x, y) on the unit circle, the y-coordinate represents the sine of the angle formed by the radius to that point and the positive x-axis. As the angle increases, the point moves around the circle, and the y-value traces out the characteristic sine wave.

Step-by-Step Guide to Using the Sine Calculator

Inputting Your Angle
Choosing the Correct Unit
Interpreting the Result

Our Sine Calculator is designed for ease of use. Follow these simple steps to get your result quickly and accurately.

1. Enter the Angle

In the 'Angle (x)' field, type the numerical value of the angle you wish to calculate. This can be a positive, negative, or zero value.

2. Select the Unit

Use the dropdown menu to select whether your angle is in 'Degrees' or 'Radians'. This is a critical step, as sin(30°) is very different from sin(30 rad).

3. Click Calculate

Press the 'Calculate' button. The sine of your angle will be displayed in the result section. To start over with new values, simply click the 'Reset' button.

Quick Examples

For an angle of 90 degrees, enter '90' and select 'Degrees'. The result will be 1.
For an angle of π radians (approx 3.14159), enter '3.14159' and select 'Radians'. The result will be close to 0.

Real-World Applications of the Sine Function

Physics and Engineering
Signal Processing
Geography and Navigation

The sine function is not just an abstract mathematical concept; it's essential for describing many real-world phenomena.

Oscillations and Waves

In physics, sine waves are used to model phenomena like sound waves, light waves, and the oscillations of a pendulum. The amplitude, frequency, and phase of these waves are all described using variations of the sine function.

AC Circuits

Electrical engineers use the sine function extensively to describe alternating current (AC). The voltage and current in AC circuits oscillate sinusoidally, making the sine function a fundamental tool for circuit analysis.

Common Misconceptions and Correct Methods

Degrees vs. Radians Confusion
Calculator Mode Errors
Interpreting Negative Results

A common source of error in trigonometry is misunderstanding the units. Always ensure your calculator is in the correct mode (degrees or radians) to match your input. Our calculator removes this ambiguity by letting you explicitly choose the unit.

Why is sin(180°) = 0?

On the unit circle, a 180-degree angle corresponds to the point (-1, 0). Since the sine value is the y-coordinate, sin(180°) is 0. This represents a point of zero displacement in a wave cycle.

Negative Angles

A negative angle like -45° represents a clockwise rotation on the unit circle. The sine function is an odd function, which means sin(-x) = -sin(x). Therefore, sin(-45°) is equal to -sin(45°).

Mathematical Derivation and Properties

Taylor Series Expansion
Relationship with Cosine
The Pythagorean Identity

The sine function can be defined by its Taylor series expansion around 0, which is an infinite sum of terms.

The Taylor Series

sin(x) = x - x³/3! + x⁵/5! - x⁷/7! + ... This series converges for all real numbers x, where x is in radians. Calculators use efficient algorithms based on this series to compute sine values.

The Pythagorean Identity

One of the most fundamental identities in trigonometry is sin²(x) + cos²(x) = 1. This identity is derived directly from the Pythagorean theorem applied to the unit circle and is true for any angle x.

Sine of 30°

Sine of π/2 Radians