# How to Find the Period of a Trig Function

## Steps for Finding the Period

Did you know that if you are given a trig function and asked to find the period, all you have to do is to look at one particular number and make a simple calculation?

That's right! Looking at this function, you might think you have to do something complicated, but you only need to worry about one of the numbers to figure out your period.

The **period** is defined as the length of a function's cycle. Trig functions are cyclical, and when you graph them, you'll see the ups and downs of the graph and you'll see that these ups and downs keep repeating at regular intervals.

All you have to do is to follow these steps.

#### Step 1: Rewrite your function in standard form if needed.

The first step you need to take is to make sure that your function is written in **standard form**:

The trig word in the function stands for the trig function you have, either sine, cosine, tangent, or cotangent. The A stands for the **amplitude** of the function, or how high the function gets. The B value is the one you use to calculate your period. When you divide your C by your B (C / B), you get your **phase shift**. The D stands for any vertical shift the function has. The **vertical shift** is how much above or below the *x*-axis the function is shifted.

The trig function from the beginning of this lesson, f(*x*) = 3 sin(4*x* + 2), already happens to be in standard form, so you don't have to do anything here.

#### Step 2: Label your A, B, C, and D values.

After rewriting your function in standard form if needed, now you can label your A, B, C, and D values.

For our example trig function, your A is 3, your B is 4, your C is -2, and your D is 0. Be careful here when it comes to labeling your C value. The C value is negative in the standard form, so if your C value is being added, then your C value is really negative.

#### Step 3: Calculate your period.

Your next step is to calculate your period using just the B value that you labeled in step two. You'll use two formulas to find your period.

If your trig function is either a **sine** or a **cosine**, you'll need to divide two pi by the absolute value of your B.

If your trig function is either a **tangent** or **cotangent**, then you'll need to divide pi by the absolute value of your B.

Our function, f(*x*) = 3 sin(4*x* + 2), is a sine function, so the period would be 2 pi divided by 4, our B value.

## Solution

Doing the calculation, you see that the period for the function is pi over 2, or 1.57.

Usually, when working with trig functions, you'll leave the pi as is and simplify the rest. So your answer will be pi over 2 instead of the 1.57.

You can also find the period of a trig function from its graph. If we graph out the function *y* = 3 sin(4*x* + 2), we get this graph:

The period is the distance between each repeating wave of the function, so from tip to tip of the function's graph. As you can see from this graph, the distance between the tips of the function is 3.034 - 1.463 = 1.57.

1.57 is the same as pi over 2, which is the same as we got from using the formula.

## Example

Let's look at another example.

Find the period of this function.

Looking at this function, you see that it uses the tangent function. Remembering the period formulas for the tangent function, the period is found by dividing pi by the absolute value of the B value, in this case, 2.

So, for this tangent trig function, the period is pi over 2, or half a pi.

## Lesson Summary

In order to find the **period**, or length of a function's cycle, for a trig function, there are three steps to follow:

- Put the equation into the
**standard form** - Label your values
- Calculate the period using the appropriate form based on your trig function

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## Finding the Period of Secant and Cosecant

In the video lesson, we learned how to find the period of various trigonometric functions. If the trigonometric function is in standard form:

## Definition and Graphs of Secant and Cosecant

The cosecant function is defined as one divided by sine and the secant function is defined as one divided by cosine.

The cosecant function has vertical asymptotes where the sine function is zero and the secant function has vertical asymptotes where the cosine function is zero.

Since secant and cosecant are related to sine and cosine, you may think that the period is calculated similarly to the period of sine and cosine - and you'd be right! If the secant or cosecant function is in standard form, the period of the function is given by:

## Examples

1) Find the period of f(x) = 3sec(2x-5)+3

2) Find the period of g(x) = -2csc(3x+9) - 1

3) Find the period of h(x) = 5sec(-0.5x+1)-12

## Solutions

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