# Sine Function Calculator

Created by Davide Borchia
Reviewed by Anna Szczepanek, PhD
Last updated: Jan 21, 2023

Use the sine function calculator to find and calculate the value of the sin function, calculate its graph, and much more. Keep reading to learn:

• What is the sine function;
• How to plot the calculated sine function in a graph;
• The properties of the sine function;

and more!

## Calculate the sine function: what is the sine function

The sine function is a fundamental trigonometric function corresponding to the vertical projection of the unitary radius of a circle when the radius is at a given angle from the horizontal axis. You can clearly visualize this definition in the image below. On the unit circle (i.e., with radius $r=1$), the sine oscillates between $-1$ and $+1$: we say that the function is periodic. The function returns to the same value every $360\degree$.

## Calculate the graph of the sine function

To calculate the graph of the sine function simply plug the values of the angle in the sine function calculator, and connect the points with a line. To appreciate the behavior of the sine function, start by plotting the sine function for the values $0\degree$, $90\degree$, $180\degree$, and $360\degree$. Even with just four points, you can see the oscillating behavior. Double the points by adding the multiples of $45\degree$, and you'll start to see some smoothing. Next up, half angles of $45\degree$. At this stage, from a distance, you would not notice that we are still seeing straight lines! Increasing the points makes the curve smooth: finally, we calculated our sine function graph. In the picture below, we repeated the graph of the sine function slightly more than five times: this way, you can fully appreciate the periodicity of the function. ## Other calculators for the sine function

If the sine function calculator sparked your interest, visit our other tools dedicated to this trigonometric function:

• The sin calculator;
• The sin degrees calculator; and
• The sin theta calculator.

## FAQ

### How do I calculate the graph of the sine function?

To calculate the graph of the sine function follows a few simple steps:

1. Select the angles you will use or the spacing between them. For example, if you want to plot every 45 degrees, choose: 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°.
2. Calculate the sine function values at these angles. In our case, we would have: 0, √2/2, 1, √2/2, 0, -√2/2, -1, -√2/2, and 0.
3. Plot the pairs of angles and values, and connect the points.

Increase the number of points to obtain a smoother curve.

### What is the periodicity of the sine function?

The periodicity of the sine function is 360°, meaning that the values defined for the range [0,360) repeat indefinitely. You can write this property as sin(α) = sin(α +360°). This periodicity stems from the fact that the trigonometric functions are defined on a circle.

### What is the value of the sine function for the most important angles?

For the angles 0°, 30°, 45°, 60°, and 90°, we find neat values of the sine function:

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

To find the following values, (120°, 135°, and so on) simply **mirror the values from 1 to 0.

Davide Borchia
Angle
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