# Sin Degrees Calculator

Created by Davide Borchia
Reviewed by Luis Hoyos
Last updated: Jan 18, 2024

The sin degrees calculator will teach you how to calculate and understand the sine function when its argument is an angle in degrees. Keep reading this short article to learn:

• What is the sine function;
• How to calculate the sin function in degrees;
• Some values of the sine function for relevant angles in degrees, such as:
• The sin of 45 degrees;
• The sin of 90 degrees;
• The sin of 30 degrees.

and much more!

## What is the sine function?

The sine function (often shortened to sin) is a trigonometric function, that is, a mathematical function that takes as the argument an angle and returns a numeric value. The sine function corresponds to the projection of the radius of a circle on the vertical axis as a function of the angle comprised between the radius and the positive portion of the horizontal axis.

Since the sine function takes angles as its argument, we have a few choices. We can:

• Calculate the sin function in radians (with a radian being an angle with associated arc as long as the radius);
• Calculate the sin in degrees;
• Calculate the sin in gradians, turns, or other more exotic measurement units for the angle; or
• Calculate the sin in degrees, minutes, and seconds.

## Calculate the sin in degrees: sine function for angle in degrees

In this section, you will learn to calculate the sin in degrees. When you pass an angle in degrees as the argument of the sine function, you pass a value between $0\degree$ and $360\degree$. This range is what we call the period of the sine function: the values assumed by the sine in this interval are repeated regularly outside of it. The values assumed by the sine function in a quadrant are repeated in the next one, just mirrored. For the four quadrants of the goniometric circle, we define four behaviors:

• For the first quadrant (from $0\degree$ to $90\degree$), we find values of the sine from $0$ to $1$.
• For the second quadrant (from $90\degree$ to $180\degree$), we find the values of the previous quadrant, but in the opposite order (starting from the one for $90\degree$ and ending at the value for $0\degree$).
• The third quadrant (from $180\degree$ to $270\degree$) has the same values as the first quadrant, but negative.
• The fourth quadrant (from $270\degree$ to $360\degree$) contains the same values as the second quadrant, changed in sign.

The value of the sine oscillates between $1$ and $-1$, these values included in the range. Let's see the values of the sine for the most common angles.

## Some examples: the sin of 30 degrees, the sin of 60, and many more

Let's consider some trivial angles first:

• $\sin(0\degree) = 0$;
• $\sin(90\degree) =1$.

Between those angles, we can find all possible values of the sine function.
The next angle in degrees we will consider is $45\degree$. For this angle, we imagine building a square with diagonal $1$. The diagonal splits the square in two $45\degree$-$45\degree$-$90\degree$ right triangles. We can then find the length of the sides of such triangles, and the result will also be the sine of $45\degree$:

• $\sin(45\degree) = \sqrt{2}/2$.

Other interesting angles are $30\degree$ and $60\degree$, as they appear in other special right triangles. For these angles, we have the sine of 30 and the sine of 60 degrees

• $\sin(30\degree) = 1/2$
• $\sin(60\degree) = \sqrt{3}/2$

Most other angles have more complex expressions (if they can be expressed with radicals at all), and the search for an exact value usually doesn't justify the time and the effort. Find their values with calculators like Omni's sin degree calculator.

We developed a small set of tools focused on the trigonometric function sine:

## FAQ

### What is the sine of 45 degrees?

The value of the sine of 45° can be calculated with the following steps:

1. Consider a circle with a radius of one.
2. Trace the radius at 45° about the horizontal axis and trace two orthogonal lines departing from its intersection with the circumference. Note that you draw a square with diagonal d = 1.
3. The horizontal and vertical sides of the squares are, respectively, the cosine and the sine of the angle 45°. To calculate them, use the formula for the diagonal of a court: d = l · √2.
4. Find the sine: sin(45°) = 1/√2 = √2/2

### What are the values of the sine in degrees?

The sine values in degrees oscillate from -1 to +1. The angles between 0° and 90° have positive values starting from 0 and ending at +1. All values of the sine in degrees repeat cyclically. You can calculate them with the following relationships:

• sin(α + 90°) = sin(90° - α);
• sin(α + 180°) = -sin(α); and
• sin(α + 270°) = -sin(90° - α)

### What is the sine of an angle?

The sine of an angle is the measure of the vertical projection of the radius of a circle placed at that angle with respect to the horizontal axis. The sine also defines one of the legs of a right triangle, where one of the acute angles is the angle of which you are calculating the sine.

Davide Borchia
Angle
deg
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