# Graphing Trigonometric Functions

Certainly, the best way to understand the behavior of trigonometric functions is to create a visual representation of their graphs on the coordinate plane. This helps us to identify their key features and to analyze the impact of these features on the appearance of each graph. However, do you know what steps to follow to graph trigonometric functions and their reciprocal functions? If your answer is no, then do not worry, as we will guide you through the process.

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In this article, we will define what graphs of trigonometric functions are, discuss their key features, and we will show you how to graph trigonometric functions and their reciprocal functions using practical examples.

Graphs of trigonometric functions are graphical representations of functions or ratios defined based on the sides and the angles of a right triangle. These include the functions sine (sin), cosine (cos), tangent (tan), and their corresponding reciprocal functions cosecant (csc), secant (sec) and cotangent (cot).

## What are the key features of trigonometric functions graphs?

Before we go through the process to graph trigonometric functions, we need to identify some key features about them:

### Amplitude

The amplitude of trigonometric functions refers to the vertical stretch factor, which you can calculate as the absolute value of half the difference between its maximum value and its minimum value.

The amplitude of the functions $y=\mathrm{sin}\theta$ and $y=\mathrm{cos}\theta$ is $\left|\frac{1-\left(-1\right)}{2}\right|=1$.

For functions in the form $y={\mathbit{a}}\mathrm{sin}b\theta$, or $y={\mathbit{a}}\mathrm{cos}b\theta$, the amplitude equals the absolute value of a.

$Amplitude=\left|a\right|$

If you have the trigonometric function $y={\mathbf{2}}\mathrm{sin}\theta$, then the amplitude of the function is 2.

The tangent functions graph has no amplitude, as it does not have a minimum or maximum value.

### Period

The period of trigonometric functions is the distance along the x-axis from where the pattern starts, to the point where it starts again.

The period of sine and cosine is 2π or 360º.

For functions in the form $y=a\mathrm{sin}{\mathbit{b}}\theta$, or $y=a\mathrm{cos}{\mathbit{b}}\theta$, b is known as the horizontal stretch factor, and you can calculate the period as follows:

$Period=\frac{2\mathrm{\pi }}{\left|\mathrm{b}\right|}or\frac{360°}{\left|b\right|}$

For functions in the form $y=a\mathrm{tan}{\mathbit{b}}\theta$, the period is calculated like this:

$Period=\frac{\mathrm{\pi }}{\left|b\right|}or\frac{180°}{\left|b\right|}$

Find the period of the following trigonometric functions:

• $y=cos\frac{\mathrm{\pi }}{2}\theta$
• $y=\mathrm{tan}\frac{1}{3}\theta$

### Domain and range

The domain and range of the main trigonometric functions are as follows:

 Trigonometric function Domain Range Sine All real numbers $-1\le y\le 1$ Cosine All real numbers $-1\le y\le 1$ Tangent All real numbers, apart from$\frac{\mathrm{n\pi }}{2},wheren=±1,±3,±5,...$ All real numbers Cosecant All real numbers, apart from $n\mathrm{\pi },\mathrm{where}\mathrm{n}=0,±1,±2,±3,...$ $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ Secant All real numbers, apart from $\frac{\mathrm{n\pi }}{2},wheren=±1,±3,±5,...$ $\left(-\infty ,-1\right]\cup \left[1,\infty \right)$ Cotangent All real numbers, apart from $n\mathrm{\pi },\mathrm{where}\mathrm{n}=0,±1,±2,±3,...$ All real numbers

Remember that all trigonometric functions are periodic, because their values repeat over and over again after a specific period.

## How to graph trigonometric functions?

To graph the trigonometric functions you can follow these steps:

• If the trigonometric function is in the form $y=a\mathrm{sin}b\theta$, $y=a\mathrm{cos}b\theta$, or $y=a\mathrm{tan}b\theta$, then identify the values of a and b, and work out the values of the amplitude and the period as explained above.

• Create a table of ordered pairs for the points that you will include in the graph. The first value in the ordered pairs will correspond to the value of the angle θ, and the values of y will correspond to the value of the trigonometric function for the angle θ, for example, sin θ, so the ordered pair will be (θ, sin θ). The values of θ can be either in degrees or radians.

You can use the unit circle to help you work out the values of sine and cosine for the most commonly used angles. Please read about Trigonometric Functions, if you need to recap how to do this.

• Plot a few points on the coordinate plane to complete at least one period of the trigonometric function.

• Connect the points with a smooth and continuous curve.

### Sine graph

Sine is the ratio of the length of the opposite side of the right triangle over the length of the hypotenuse.

The graph for a sine function $y=\mathrm{sin}\theta$ looks like this:

Sine graph, Marilú García De Taylor - StudySmarter Originals

From this graph we can observe the key features of the sine function:

• The graph repeats every 2π radians or 360°.

• The minimum value for sine is -1.

• The maximum value for sine is 1.

• This means that the amplitude of the graph is 1 and its period is 2π (or 360°).

• The graph crosses the x-axis at 0 and every π radians before and after that.

• The sine function reaches its maximum value at π/2 and every 2π before and after that.

• The sine function reaches its minimum value at 3π/2 and every 2π before and after that.

Graph the trigonometric function $y=4\mathrm{sin}2\theta$

• Identify the values of a and b

$a=4,b=2$

• Calculate the amplitude and period:

$Amplitude=\left|a\right|=\left|4\right|=4\phantom{\rule{0ex}{0ex}}Period=\frac{2\mathrm{\pi }}{\left|\mathrm{b}\right|}=\frac{2\mathrm{\pi }}{\left|2\right|}=\frac{\overline{)2}\mathrm{\pi }}{\overline{)2}}=\mathrm{\pi }$

• Table of ordered pairs:
 θ $y=4\mathrm{sin}2\theta$ 0 0 $\frac{\mathrm{\pi }}{4}$ 4 $\frac{\mathrm{\pi }}{2}$ 0 $\frac{3\mathrm{\pi }}{4}$ -4 $\mathrm{\pi }$ 0
• Plot the points and connect them with a smooth and continuous curve:

Sine graph example, Marilú García De Taylor - StudySmarter Originals

### Cosine graph

Cosine is the ratio of the length of the adjacent side of the right triangle over the length of the hypotenuse.

The graph for the cosine function $y=\mathrm{cos}\theta$looks exactly like the sine graph, except that it is shifted to the left by π/2 radians, as shown below.

Cosine graph, Marilú García De Taylor - StudySmarter Originals

By observing this graph, we can determine the key features of the cosine function:

• The graph repeats every 2π radians or 360°.

• The minimum value for cosine is -1.

• The maximum value for cosine is 1.

• This means that the amplitude of the graph is 1 and its period is 2π (or 360°).

• The graph crosses the x-axis at π/2 and every π radians before and after that.

• The cosine function reaches its maximum value at 0 and every 2π before and after that.

• The cosine function reaches its minimum value at π and every 2π before and after that.

Graph the trigonometric function $y=2\mathrm{cos}\frac{1}{2}\theta$

• Identify the values of a and b:
$a=2,b=\frac{1}{2}$
• Calculate the amplitude and period:
• Table of ordered pairs:
 θ $y=2\mathrm{cos}\frac{1}{2}\theta$ 0 2 $\mathrm{\pi }$ 0 $2\mathrm{\pi }$ -2 $3\mathrm{\pi }$ 0 $4\mathrm{\pi }$ 2
• Plot the points and connect them with a smooth and continuous curve:

Cosine graph example, Marilú García De Taylor - StudySmarter Originals

### Tangent graph

Tangent is the ratio of the length of the opposite side of the right triangle over the length of the adjacent side.

The graph of the tangent function $y=\mathrm{tan}\theta$, however, looks a bit different than the cosine and sine functions. It is not a wave but rather a discontinuous function, with asymptotes:

Tangent graph, Marilú García De Taylor - StudySmarter Originals

By observing this graph, we can determine the key features of the tangent function:

• The graph repeats every π radians or 180°.

• No minimum value.

• No maximum value.

• This means that the tangent function has no amplitude and its period is π (or 180°).

• The graph crosses the x-axis at 0 and every π radians before and after that.

• The tangent graph has asymptotes, which are values where the function is undefined.

• These asymptotes are at π/2 and every π before and after that.

The tangent of an angle can also be found with this formula:

$\mathrm{tan}\theta =\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }$

Graph the trigonometric function $y=\frac{3}{4}\mathrm{tan}\theta$

• Identify the values of a and b:
$a=\frac{3}{4},b=1$

• Calculate the amplitude and period:
Tangent functions have no amplitude.
• Table of ordered pairs:
 θ $y=\frac{3}{4}\mathrm{tan}\theta$ $-\frac{\mathrm{\pi }}{2}$ undefined(asymptote) $-\frac{\mathrm{\pi }}{4}$ $-\frac{3}{4}$ 0 0 $\frac{\mathrm{\pi }}{4}$ $\frac{3}{4}$ $\frac{\mathrm{\pi }}{2}$ undefined(asymptote)
• Plot the points and connect them:

Tangent graph example, Marilú García De Taylor - StudySmarter Originals

## What are the graphs of the reciprocal trigonometric functions?

Each trigonometric function has a corresponding reciprocal function:

• Cosecant is the reciprocal of sine.
• Secant is the reciprocal of cosine.
• Cotangent is the reciprocal of tangent.

To graph the reciprocal trigonometric functions you can proceed as follows:

### Cosecant graph

The graph of the cosecant function $y=csc\theta$ can be obtained like this:

• Graph the corresponding sine function first, to use it as a guide.
• Draw vertical asymptotes in all the points where the sine function intercepts the x-axis.
• The cosecant graph will touch the sine function at its maximum and minimum value. From those points, draw the reflection of the sine function, which approaches but never touches the vertical asymptotes and extends to positive and negative infinity.

Cosecant graph, Marilú García De Taylor - StudySmarter Originals

The cosecant function graph has the same period as the sine graph, which is 2π or 360°, and it has no amplitude.

Graph the reciprocal trigonometric function $y=2csc\theta$

• $a=2,b=1$
• No amplitude
• $Period=\frac{2\mathrm{\pi }}{\left|\mathrm{b}\right|}=\frac{2\mathrm{\pi }}{\left|1\right|}=\frac{2\mathrm{\pi }}{1}=2\mathrm{\pi }$

Cosecant graph example, Marilú García De Taylor - StudySmarter Originals

### Secant graph

To graph the secant function $y=sec\theta$ you can follow the same steps as before, but using the corresponding cosine function as a guide. The secant graph looks like this:

Secant graph, Marilú García De Taylor - StudySmarter Originals

The secant function graph has the same period as the cosine graph, which is 2π or 360°, and it also has no amplitude.

Graph the reciprocal trigonometric function $y=\frac{1}{2}sec2\theta$

• $a=\frac{1}{2},b=2$
• No amplitude
• $Period=\frac{2\mathrm{\pi }}{\left|\mathrm{b}\right|}=\frac{2\mathrm{\pi }}{\left|2\right|}=\frac{\overline{)2}\mathrm{\pi }}{\overline{)2}}=\mathrm{\pi }$

Secant graph example, Marilú García De Taylor - StudySmarter Originals

### Cotangent graph

The cotangent graph is very similar to the graph of tangent, but instead of being an increasing function, cotangent is a decreasing function. The cotangent graph will have asymptotes in all the points where the tangent function intercepts the x-axis.

Cotangent graph, Marilú García De Taylor - StudySmarter Originals

The period of the cotangent graph is the same as the period of the tangent graph, π radians or 180°, and it also has no amplitude.

Graph the reciprocal trigonometric function $y=3cot\theta$

• $a=3,b=1$
• No amplitude
• $Period=\frac{\mathrm{\pi }}{\left|\mathrm{b}\right|}=\frac{\mathrm{\pi }}{\left|1\right|}=\frac{\mathrm{\pi }}{1}=\mathrm{\pi }$

Cotangent graph example, Marilú García De Taylor - StudySmarter Originals

## What are the graphs of the inverse trigonometric functions?

The inverse trigonometric functions refer to the arcsine, arccosine and arctangent functions, which can also be written as $S{\mathrm{in}}^{-1},C{\mathrm{os}}^{-1}$ and $T{\mathrm{an}}^{-1}$. These functions do the opposite of the sine, cosine and tangent functions, which means that they give back an angle when we plug a sin, cos or tan value into them.

Remember that the inverse of a function is obtained by swapping x and y, that is, x becomes y and y becomes x.

The inverse of $y=\mathrm{sin}x$ is $x=\mathrm{sin}y$, and you can see its graph below:

Inverse of sine graph, Marilú García De Taylor - StudySmarter Originals

However, in order to make the inverses of trigonometric functions become functions, we need to restrict their domain. Otherwise, the inverses are not functions because they do not pass the vertical line test. The values in the restricted domains of the trigonometric functions are known as principal values, and to identify that these functions have a restricted domain, we use capital letters:

 Trigonometric function Restricted domain notation Principal values Sine $y=Sinx$ $-\frac{\mathrm{\pi }}{2}\le x\le \frac{\mathrm{\pi }}{2}$ Cosine $y=Cosx$ $0\le x\le \mathrm{\pi }$ Tangent $y=Tanx$ $-\frac{\mathrm{\pi }}{2}

### Arcsine graph

Arcsine is the inverse of the sine function. The inverse of $y=Sinx$ is defined as $x=Si{n}^{-1}y$ or $x=Arc\mathrm{sin}y$. The domain of the arcsine function will be all real numbers from -1 to 1, and its range is the set of angle measures from $-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$. The graph of the arcsine function looks like this:

Arcsine graph, Marilú García De Taylor - StudySmarter Originals

### Arccosine graph

Arccosine is the inverse of the cosine function. The inverse of $y=Cosx$ is defined as $x=Co{s}^{-1}y$ or $x=Arc\mathrm{cos}y$. The domain of the arccosine function will be also all real numbers from -1 to 1, and its range is the set of angle measures from $0\le y\le \mathrm{\pi }$. The graph of the arccosine function is shown below:

Arccosine graph, Marilú García De Taylor - StudySmarter Originals

### Arctangent graph

Arctangent is the inverse of the tangent function. The inverse of $y=Tanx$ is defined as$x=Ta{n}^{-1}y$ or $x=Arc\mathrm{tan}y$. The domain of the arctangent function will be all real numbers, and its range is the set of angle measures between $-\frac{\mathrm{\pi }}{2}. The arctangent graph looks like this:

Arctangent graph, Marilú García De Taylor - StudySmarter Originals

If we graph all the inverse functions together, they look like this:

Arcsine, Arccosine, and Arctangent graphs together, Marilú García De Taylor - StudySmarter Originals

## Graphing trigonometric functions - Key takeaways

• Graphs of trigonometric functions are graphical representations of functions or ratios defined based on the sides and the angles of a right triangle.
• The key features of trigonometric functions are: amplitude, period, domain and range.
• The amplitude of trigonometric functions refers to the vertical stretch factor, which you can calculate as the absolute value of half the difference between its maximum value and its minimum value.
• The period of trigonometric functions is the distance along the x-axis from where the pattern starts, to the point where it starts again.
• Each trigonometric function has a corresponding reciprocal function. Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent.
• The inverse trigonometric functions arcsine, arccosine and arctangent, do the opposite of the sine, cosine and tangent functions, which means that they give back an angle when we plug a sin, cos or tan value into them.

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What are graphs of trigonometric functions?

Graphs of trigonometric functions are graphical representations of functions or ratios defined based on the sides and the angles of a right triangle. These include the functions sine (sin), cosine (cos), tangent (tan), and their corresponding reciprocal functions cosecant (csc), secant (sec) and cotangent (cot).

What are the rules when graphing trigonometric functions?

• Identify its key features: amplitude (vertical stretch factor) and period.
• Plot a few points on the coordinate plane to complete one period of the function.
• Connect the points with a smooth and continuous curve.
• Continue the graph if required, by repeating the pattern after each period.

How to graph trigonometric functions?

To graph the trigonometric functions you can follow these steps:

• If the trigonometric function is in the form y = a sin bθ, y = a cos bθ, or y = a tan bθ, then identify the values of a and b, and work out the values of the amplitude and the period.
• Create a table of ordered pairs for the points to include in the graph. The first value in the ordered pairs will correspond to the value of the angle θ, and the values of y will correspond to the value of the trigonometric function for the angle θ, for example, sin θ, so the ordered pair will be (θ, sin θ). The values of θ can be either in degrees or radians.
• Plot a few points on the coordinate plane to complete at least one period of the trigonometric function.
• Connect the points with a smooth and continuous curve.

What is an example of trigonometric function graphs?

The graph for a sine function has the following characteristics:

• It has a wave shape.
• The graph repeats every 2π radians or 360°.
• The minimum value for sine is -1.
• The maximum value for sine is 1.
• This means that the amplitude of the graph is 1 and its period is 2π (or 360°).
• The graph crosses the x-axis at 0 and every π radians before and after that.

How to draw graphs of inverse trigonometric functions?

To draw graphs of inverse trigonometric functions proceed as follows:

• Restrict the domain of the trigonometric function to its principal values.
• Work out the domain and range. The domain of the inverse will be the range of its corresponding trigonometric function, and the range of the inverse will be the restricted domain of its trigonometric function.
• Plot a few points and connect them with a smooth and continuous curve.

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