# Period of the Sine Function – Formulas and Examples

The period of the sine function is 2π. This means that the value of the function is the same every 2π units. Similar to other trigonometric functions, the sine function is a periodic function, which means that it repeats at regular intervals. The interval of the sine function is 2π.

For example, we have sin(π) = 0. If we add 2π to the input of the function, we have sin(π + 2π), which is equal to sin(3π). Since we have sin(π) = 0, we also have sin(3π) = 0. Every time we add 2π of the input values, we will get the same result.

##### TRIGONOMETRY

Relevant for

Learning to find the period of the sine function.

See examples

##### TRIGONOMETRY

Relevant for

Learning to find the period of the sine function.

See examples

## Period of the basic sine function

The basic sine function is $latex y = \sin(x)$. Since this function can be evaluated for any real number, the sine function is defined for all real numbers. The period of this function can be clearly observed in its graph since it is the distance between “equivalent” points.

Since the graph of $latex y = \sin(x)$ looks like a single pattern that repeats itself over and over, we can think of the period as the distance on the x-axis before the graph starts repeating.

Looking at the graph, we see that the graph repeats itself after 2π. This means that the function is periodic with a period of 2π. In the unit circle, 2π equals one complete revolution around the circle.

Any quantity greater than 2π means that we are repeating the revolution. This is the reason why the value of the function is the same every 2π.

## Changing the period of the sine function

The period of the basic sine function $latex y = \sin (x)$ is 2π, but if x is multiplied by a constant, the period of the function can change.

If x is multiplied by a number greater than 1, that “speeds up” the function and the period will be smaller. That means it won’t take long for the function to start repeating itself.

For example, if we have the function $latex y = \sin(2x)$, this causes the “speed” of the original function to double. In this case, the period is π.

On the other hand, if x is multiplied by a number between 0 and 1, this causes the function to slow down and the period will be larger since it will take longer for the function to repeat itself. For example, the function $latex y = \sin (\frac{x}{2})$ halves the “speed” of the original function. The period of this function is 4π.

## Finding the period of a sine function

To find the period of a sine function, we have to consider the coefficient of x that is inside the function. We can use B to represent this coefficient. Therefore, if we have an equation in the form $latex y = \sin(Bx)$, we have the following formula:

In the denominator, we have |B|. This means that we take the absolute value of B. Thus, if B is a negative number, we just take the positive version of the number.

This formula works even if we have more complex variations of the sine function like $latex y = 3 \sin(2x + 4)$. Only the coefficient of x matters when calculating the period, so we would have:

$latex \text{Period}=\frac{2\pi}{|B|}$

$latex \text{Period}=\frac{2\pi}{2}$

$latex \text{Period}=\pi$

## Period of the sine function – Examples with answers

What you have learned about the period of sine functions is used to solve the following examples. Try to solve the problems yourself before looking at the answer.

### EXAMPLE 1

What is the period of the function $latex y = \sin(3x)$?

We use the period formula with the value $latex | B | = 3$. Therefore, we have:

$latex \text{Period}=\frac{2\pi}{|B|}$

$latex \text{Period}=\frac{2\pi}{3}$

The period of the function is $latex \frac{2}{3}\pi$.

### EXAMPLE 2

We have the sine function $latex y = 3 \sin(4x)+1$. What is its period?

The only value we need is the coefficient of x. Therefore, we use the value $latex | B | = 4$ in the formula for the period:

$latex \text{Period}=\frac{2\pi}{|B|}$

$latex \text{Period}=\frac{2\pi}{4}$

$latex \text{Period}=\frac{\pi}{2}$

The period of the function is $latex \frac{\pi}{2}$.

### EXAMPLE 3

What is the period of the function $latex y = \frac{1}{2} \sin (- \frac{1}{4} x-4)$?

We just have to use the coefficient of x to find the period. We see that in this case, the coefficient is negative, so we take its positive version. Therefore, we use the value $latex |B| = \frac{1}{4}$ in the period formula:

$latex \text{Period}=\frac{2\pi}{|B|}$

$latex \text{Period}=\frac{2\pi}{\frac{1}{4}}$

$latex \text{Period}=8\pi$

The period of the function is $latex 8\pi$.

## Period of the sine – Practice problems

Solve the following practice problems using what you have learned about the period of sine functions. If you need help with this, you can look at the solved examples above.