What Is The Period Of Sine Function?

The period of the sine function is ​2π​, which means that the value of the function is the same every 2π units.

The sine function, like cosine, tangent, cotangent, and many other trigonometric function, is a ​periodic function​, which means it repeats its values on regular intervals, or "periods." In the case of the sine function, that interval is 2π.

For instance, sin(π) = 0. If you add 2π to the ​x​-value, you get sin(π + 2π), which is sin(3π). Just like sin(π), sin(3π) = 0. Every time you add or subtract 2π from our ​x​-value, the solution will be the same.

You can easily see the period on a graph, as the distance between "matching" points. Since the graph of ​y​ = sin(​x​) looks like a single pattern repeated over and over again, you can also think of it as the distance along the ​x​-axis before the graph starts to repeat itself.

On the unit circle, 2π is a trip all the way around the circle. Any amount greater than 2π radians means that you keep looping around the circle – that's the repeating nature of the sine function, and another way to illustrate that every 2π units, the function's value will be the same.

The period of the basic sine function

\(y = \sin(x)\)

is 2π, but if ​x​ is multiplied by a constant, that can change the value of the period.

If ​x​ is multiplied by a number greater than 1, that "speeds up" the function, and the period will be smaller. It won't take as long for the function to start repeating itself.

For example,

\(y = \sin(2x)\)

doubles the "speed" of the function. The period is only π radians.

But if ​x​ is multiplied by a fraction between 0 and 1, that "slows down" the function, and period is larger because it takes a longer time for the function to repeat itself.

For example,

\(y = \sin\bigg(\frac{x}{2} \bigg)\)

cuts the "speed" of the function in half; it takes a long time (4π radians) for it to complete a full cycle and start to repeat itself again.

Say you want to calculate the period of a modified sine function like

\(y = \sin(2x) \text{ or } y = \sin\bigg(\frac{x}{2}\bigg)\)

The coefficient of ​x​ is the key; let's call that coefficient ​B​.

So if you have an equation in the form ​y​ = sin(​Bx​), then:

\(\text{Period} = \frac{2π}{|B|}\)

The bars | | mean "absolute value," so if ​B​ is a negative number, you would just use the positive version. If ​B​ was −3, for instance, you would just go with 3.

This formula works even if you have a complicated-looking variation of the sine function, like

\(y = \frac{1}{3}× \sin(4x + 3)\)

The coefficient of ​x​ is all that matters for calculating the period, so you would still do:

\(\text{Period} = \frac{2π}{|4|}\)
\(\,\)
\(\text{Period} = \frac{π}{2}\)

To find the period of cosine, tangent and other trig functions, you use a very similar process. Just use the standard period for the specific function you're working with when you calculate.

Since the period of cosine is 2π, the same as sine, the formula for the period of a cosine function will be the same as it is for sine. But for other trig functions with a different period, like tangent or cotangent, we make a slight adjustment. For example, the period of cot(​x​) is π, so the formula for the period of ​y​ = cot(3​x​) is:

\(\text{Period} = \frac{π}{|3|}\)

where we use π instead of 2π.

\(\text{Period} = \frac{π}{3}\)