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High School Precalculus: Help and Review32 chapters | 297 lessons

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Lesson Transcript

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we explore how to determine the frequency of trigonometric functions. In addition to the sine function, we will look at cosine, tangent, cotangent, cosecant, and secant.

**Frequency** is a measure of how often something repeats. **Trigonometric functions** are repetitive. The trigonometric functions we'll look at today include sine, cosine, tangent, cotangent, secant, and cosecant. We'll plot the function and then measure the **period**, which is the distance between two identical and consecutive portions of a curve. Then we'll take the reciprocal of the period to obtain the frequency.

Let's break down the process of finding the frequency into five simple steps.

For this lesson, we'll assume we already have the plot. For example, the plot of *y* = sin(Ï€*t*) looks like this:

The green dashed vertical lines are described in the next step.

For the sine function, we can spot the locations of the maximum portion of the curve. In the plot, the vertical lines locate two consecutive maximums.

The crossing of the vertical line on the horizontal axis tells us a value. The green vertical line crosses the horizontal axis at *x* = 0.5 and at *x* = 2.5. The period, *T*, is the difference between these two values.

We will assume the units of the horizontal axis are seconds. Thus, *T* = 2 seconds.

The frequency, *f*, is the reciprocal of the period. Thus,

The units of frequency are cycles per second, which are also called **hertz**. Thus, the frequency is 0.5 hertz.

A frequency of 0.5 hertz is the same as 1/2 hertz. Interpreting this result is a good way to check our understanding and verify we're consistent with the graph.

1/2 hertz indicates one cycle every two seconds. And this is exactly what we see on the graph.

We can use the same method to determine the frequency of other trig functions. Let's work through a few examples.

First, we'll look at the **cosine function**. The cosine function is a shifted version of the sine function.

The minimums were chosen as the identifiable feature here, but any easily recognizable feature will work. The period is the difference between 3 seconds and 1 second. Thus, *T* = 3 - 1 = 2 seconds. The frequency is 1/*T*. Thus, *f* = 1/2 hertz.

Next, we'll look at the **tangent** function. The tangent is the sine divided by the cosine. This function is also repetitive so we can find its frequency. The equation which is plotted is *y* = tan(Ï€*t*/2).

The place where the curve crosses the horizontal axis is called the **zero-crossing**. This is an easy feature to identify. The period, *T* = 4 - 2 = 2 seconds. Thus, the frequency, *f* = 1/*T* = 1/2 hertz.

Now let's find the frequency of the **cotangent** function. Cotangent is the reciprocal of tangent. Thus, cotangent is the cosine divided by the sine. The function plotted is *y* = cot(Ï€*t*/2).

The period is 3 - 1 = 2 seconds. Thus, the frequency is 1/2 hertz.

Next is the **cosecant function**. The cosecant function is the reciprocal of the sine function. Plotted is *y* =csc(Ï€*t*).

Taking two consecutive features, we get *T* = 2.5 - 0.5 = 2 seconds. Thus, *f* = 1/2 hertz.

And, finally, let's find the frequency of the **secant** function. The secant is 1/cosine. The function plotted is *y* = sec(Ï€*t*).

*T* = 4 - 2 = 2 seconds. *f* = 1/2 hertz.

**Frequency** is a measure of how often something repeats. In this lesson, we looked at five simple steps to find the frequency of trigonometric functions. These steps include:

- Plot the function
- Identify a repeating part of the curve
- Measure the
**period**, which is the distance between two identical and consecutive portions of the curve - Calculate the frequency
- Interpret the result

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High School Precalculus: Help and Review32 chapters | 297 lessons

- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Graphing the Cosecant, Secant & Cotangent Functions 7:10
- Using Graphs to Determine Trigonometric Identity 5:02
- Solving a Trigonometric Equation Graphically 5:45
- How to Graph cos(x) 5:10
- How to Graph 1-cos(x) 6:15
- How to Find the Period of a Trig Function 4:19
- How to Find the Phase Shift of a Trig Function
- How to Find the Vertical Shift of a Trig Function
- How to Find the Frequency of a Trig Function 4:59
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