Graphs of Trigonometric Functions: The graphs of trigonometric functions are one of the most widely used tools in Science and Engineering. You may have seen professionals do various calculations for specific tasks even without realising the involvement of trigonometric graphs there. For example, sine is used to precisely locate tumours inside the brain and cosine is widely used by surveyors to identify an unknown distance.

Thus, trigonometry is being used by people in different jobs in various other fields such as medical and astronomy. In this article, let us aim to learn how to plot the graphs of trigonometric functions that are periodic, meaning they repeat themselves after a specific time period.

Trigonometric Functions

We know that there are \(6\) basic trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. These are defined as ratios of the sides of a right triangle and are also called trigonometric ratios.

As you can see, the trigonometric functions are defined in the domain \(\theta\), which is usually in radians or degrees.

Values of Trigonometric Functions

The principal values of the six trigonometric ratios are derived from the unit circle. Some of them are shared in the table given below.

These are also called the standard values of trigonometric functions for specific values of \(\theta\).

But what happens when you extrapolate these values from \(90^{\circ}\) to \(360^{\circ} ?\)

These values of trigonometric ratios in the range \(0\) to \(2 \pi\) fall in the four quadrants, as shown below.

We can find a specific pattern for each trigonometric function distributed across the four quadrants by observing the image above. The quickest way to remember this is the CAST rule. It is described in the image below.

Values in Four Quadrants

The altering of trigonometric ratios for different values of \(\theta\) in different quadrants is as tabulated below.

How to Graph Trigonometric Functions?

Let us begin with the sine function.

Step 1: Using the standard values of trigonometric functions, we plot the sine curve, as shown below.

Here, \(x^{2}+y^{2}=1\) is the unit circle in the \(xy\)- plane whose radius is \(1\) unit.

Step 2: Since we have plotted up to \(\pi\), using the CAST rule, we know that the values of the sine function are negative from \(\pi\) to \(2 \pi\).

\(\sin \theta = \left\{ {\begin{array}{*{20}{c}} { – \sin \theta ,\,{\text{for}}\,\pi {\text{ + }}\theta {\text{,}}\,{\text{2}}\pi {\text{ – }}\theta } \\ { – \cos \theta ,\,{\text{for}}\,\frac{{3\pi }}{2} \pm \theta } \end{array}} \right.\)

Step 3: Plotting this, the graph of sine function from \(0\) to \(2 \pi\) looks as shown below.

Step 4: As trigonometric functions are periodic with a time period of \(2 \pi\) or \(360^{\circ}\), the sine graph for the interval of \([-2 \pi, 2 \pi]\) can be drawn as:

Graphs of Other Trigonometric Functions

As we have already learned how to draw the sine function graph, let us see the graphs of the other trigonometric functions in this section.

1. Graph of the cosine function

2. Graph of the tangent function

3. Graph of the cotangent function

4. Graph of the secant function

5. Graph of the cosecant function

Characteristics of Sine and Cosine Functions

The sine and cosine curves are similar in many ways and yet have distinct characteristics too. Some of them are listed below.

No.	Characteristics	Sine	Cosine
1.	Graph of the function	Periodic function	Periodic function
2.	Time period	\(2 \pi\)	\(2 \pi\)
3.	Domain	\((-\infty, \infty)\)	\((-\infty, \infty)\)
4.	Range	\([-1,1]\)	\([-1,1]\)
5.	\(x-\) intercept	\(x=k \pi, \forall k\)	\(x=\frac{\pi}{2}+k \pi, \forall k\)
6.	\(y-\) intercept	\(y=0\)	\(y=1\)
7.	Type of function (odd/even)	Odd	Even
8.	Line of symmetry	Origin	\(y-\) axis

Characteristics of Tangent and Cotangent Functions

No.	Characteristics	Tangent	Cotangent
1.	Graph of the function	Periodic function	Periodic function
2.	Time period	\(\pi\)	\(\pi\)
3.	Domain	Excludes \(\frac{\pi}{2}+k \pi\), for all \(k\)	Excludes \(k \pi\), for all \(k\)
4.	Range	\([-\infty, \infty]\)	\([-\infty, \infty]\)
5.	\(x-\) intercept	\(x=k \pi, \forall k\)	\(x=\frac{\pi}{2}+k \pi, \forall k\)
6.	\(y-\) intercept	\(y=0\)	Not applicable
7.	Type of function(odd\even)	Odd	Odd
8.	Line of symmetry	Origin	Origin
9.	Vertical asymptotes	\(x=\frac{\pi}{2}+k \pi\)	\(x=k \pi\)

Characteristics of Secant and Cosecant Functions

No.	Characteristics	Tangent	Cotangent
1.	Graph of the function	Periodic function	Periodic function
2.	Time period	\(2 \pi\)	\(2 \pi\)
3.	Domain	Excludes \(\frac{\pi}{2}+k \pi\), for all \(k\)	Excludes \(k \pi\), for all \(k\)
4.	Range	\((-\infty,-1] \cup[1, \infty)\)	\((-\infty,-1] \cup[1, \infty)\)
5.	\(x-\) intercept	Not applicable	Not applicable
6.	\(y-\) intercept	\(y=1\)	Not applicable
7.	Type of function(odd\even)	Even	Odd
8.	Line of symmetry	\(y-\) axis	Origin
9.	Vertical asymptotes	\(x=\frac{\pi}{2}+k \pi\)	\(x=k \pi\)

Key Features of Trigonometric Curves

There are five key features for every trigonometric graph. They are:

• Amplitude – height of the curve from the centre line
  Vertical Shift – the displacement of the graph perpendicular to the \(x-\) axis.
• Time Period – distance between two consecutive peaks in the graph
• Phase – how far the graph is shifted from the original position
  Phase Shift – the displacement of the graph perpendicular to the \(y-\) axis.

These features may be represented on the graph, as shown below.

Graph of General Form of Functions

The general form of sine functions is given by

\(y=A \sin (B(x+C))+D\)

Here,

\(A \rightarrow\) alters the amplitude of the wave in \(y-\) direction above and below the \(x-\) axis.

\(B \rightarrow\) affects the time period and causes the phase-shift

\(C \rightarrow\) shifts the phase

\(D \rightarrow\) shifts the centre line

The graphical representation of the same is,

Amplitude Shifts

Observe that in the basic functions such as \(y=\sin x\) the value of \(A=1\). This is the value of the amplitude. Altering the value of \(A\) will stretch or compress the graph accordingly.

Observe the example of sine curves given below.

Although all the three curves are the same, their maximum and minimum vary according to the value of \(A\) in the function. For example, when \(A=3\), the maximum of the curve is \(3\), and the minimum is \(-3\).

Phase Shifts

Phase shifts of graphs are also called horizontal shifts. This shift is represented as \(y = \sin (B(x – C))\).

Note the negative sign in the equation.

Here, the phase shift is calculated as \(\frac{C}{B}\).

For example, consider \(y=\cos (3 x-\pi)\).

Here, phase shift \(=\frac{C}{B}=\frac{\pi}{3}\) (positive shift)

The graphs of \(y=\cos 3 x\) and \(y=\cos (3 x-\pi)\) are shown here. Observe to identify the shift.

But, for \(y=\cos \left(2 x+\frac{\pi}{2}\right)\), the phase shift \(=\frac{-\pi}{4}\) (negative shift)

Note:

• Negative phase shift \(\rightarrow\) graph shifts to the left or negative direction.
• Positive phase shift \(\rightarrow\) graph shifts to the right or positive direction.

Time Periods

Time period is defined as the length of one cycle in a graph. After which, they repeat themselves. As we already know, the time period of trigonometric graphs is influenced by the value of \(B\).

For sine and cosine functions, time period \(=\frac{2 \pi}{|B|}\)

For tangent and cotangent functions, time period \(=\frac{\pi}{|B|}\)

Depending on the value of \(B\), the period of graphs compresses \((B>1)\) or expands \((B<1)\).

Vertical Shifts

Vertical shift is when the entire graph is moved up or down along the vertical axis. This is denoted by \(D\) in \(y=A \sin (B(x+C))+D\).

For example, the graph of \(y=\tan x-2\) is moved down by \(2\) units.

The graph of \(y=\sin x+3\) is moved up \(3\) units.

Graphs of Inverse Trigonometric Functions

These are also called arc functions. In contrast to trigonometric functions, where we find the length of the side of a right triangle using the length of one side and one acute angle, in the case of arc functions, we find the measure of the angle using the measure of the sides. The ratios are arcsine, arccosine, arctangent, arc-cotangent, arc-secant, and arc-cosecant.

How to Draw Graphs of Arc Functions?

Let’s draw the graph for the arccosine function. It is represented as \(y=\cos ^{-1} x\).

Step 1: Draw the graph of the corresponding trigonometric function. In this case, cosine function.

Step 2: Select the portion of the graph that you want to invert.

Step 3: Draw the line \(y=x\).

Step 4: Reflect a few points in the selected portion of the trigonometric curve about the line \(y=x\).

Step 5: Join the reflected points to form the required graph of the arc function, arccosine.

Graphs of Other Arc Functions

\(y = \arcsin \,x\)	\(y = \arctan \,x\)
\(y = \operatorname{arcsec} \,x\)	\(y = \operatorname{arccot} \,x\)
\(y = {\text{arccosec}}\,x\)

Solved Problems

Q.1. Draw the graph of \(y=-\sin x\), for \(0 \leq x \leq 2 \pi\).
Sol: When a trigonometric function is multiplied by \(-1\) its graph is reflected over the \(x-\) axis.
Therefore, when \(y=\sin x\) is reflected over \(x\)-axis, we get the graph of \(y=-\sin x\).

Q.2. Calculate the amplitude, phase shift, time period, and vertical shift of the function: \(y=2 \sin x\)
Sol:
Given: \(y=2 \sin x\)
Comparing with the general form \(y=A \sin (B(x+C))+D\), we get,
\(A=2 \rightarrow\) increase amplitude by \(2 \Rightarrow\) amplitude \(=2\)
\(B=1\)
\(C=0 \Rightarrow\) There is no phase shift
\(D=0 \Rightarrow\) There is no vertical shift
Period \(=\frac{2 \pi}{|B|}=\frac{2 \pi}{1}=2 \pi\)

Q.3. Calculate the amplitude, phase shift, time period, and vertical shift of the function: \(y=\frac{1}{2} \cos (-x+\pi)-1\) and graph one period.
Sol:
Given: \(y=\frac{1}{2} \cos (-x+\pi)-1\)
Comparing with the general form \(y=A \sin (B(x+C))+D\), we get,
\(A=\frac{1}{2} \rightarrow\) decrease amplitude by half \(\Rightarrow\) amplitude \(=\frac{1}{2}\)
\(B=-1\)
\(C=-\pi\)
\(D=-1 \rightarrow\) vertical shift
Phase shift \(=\frac{C}{B}=\frac{-\pi}{-1}=\pi\)
Period \(=\frac{2 \pi}{|B|}=\frac{2 \pi}{|-1|}=2 \pi\)
The graph of \(y=\frac{1}{2} \cos (-x+\pi)-1\) for one period is,

Q.4. Graph: \(y=4 \cos 3 x+7\)
Sol:
Given: \(y=4 \cos 3 x+7\)
Comparing with the general form \(y=A \sin (B(x+C))+D\), we get,
\(A=4 \rightarrow\) increase amplitude by \(4 \Rightarrow\) amplitude \(=4\)
\(B=3\)
\(C=0 \rightarrow\) There is no phase shift
\(D=7 \rightarrow\) vertical shift, upwards
Period \(=\frac{2 \pi}{|B|}=\frac{2 \pi}{|3|}=\frac{2 \pi}{3}\)
The graph of \(y=4 \cos 3 x+7\) is,

Q.5. Graph the function: \(f(x)=2 \cos ^{-1}(x-1)\)
Sol:
Given: \(f(x)=2 \cos ^{-1}(x-1)\)
Here,
\(A=2 \rightarrow\) increase amplitude by \(2 \Rightarrow\) amplitude \(=2 \pi\)
\(B=1\)
\(C=1\)
Shift \(=\frac{C}{B}=1 \rightarrow\) moves right by \(1\) unit \(\Rightarrow x\)-intercept \(=2\)

Summary

The procedures to graph trigonometric and inverse trigonometric functions are explained in detail. There are five key features of a trigonometric function, such as the amplitude, phase, time period, phase shift, and vertical shift. These key features influence or define the graphs of trigonometric functions. The various distinct characteristics of the graphs of the six basic functions are also tabulated. This helps us understand and study these graphs better. Once you learn to draw graphs of trigonometric functions, you can select a portion of the graph and reflect it along the line \(y=x\) to get the graph of the corresponding inverse trigonometric functions.

Frequently Asked Questions (FAQs)

Q.1. How do you graph trigonometric functions?
Ans: Follow these steps to graph trigonometric functions:
–Step 1: Write the given equation in the general form as \(y=A \sin (B(x+C))+D\). (Use a function same as the function given to you.)
–Step 2: Identify the different parameters such as amplitude \((A)\), phase shift \(\left(\frac{c}{B}\right)\), and vertical shift \((D)\).
–Step 3: Period is calculated as: \(\frac{2 \pi}{|B|}\) for sine and cosine functions, and \(\frac{\pi}{|B|}\) for tangent and cotangent functions.
–Step 4: Plot the graph using the above-calculated parameters. Combine the points to form the required graph.

Q.2. What are the six trigonometric graphs?
Ans: The graphs of the six basic trigonometric functions are shown below.

Q.3. What are the key features of the graphs of trigonometric functions?
Ans: Trigonometric graphs have four key features. The function is generally represented as,
\(y=A \sin (B x+C)+D\)
Here,
\(A \rightarrow\) alters the amplitude of the wave in \(y\)-direction above and below the \(x\)-axis
\(B \rightarrow\) affects the time period and causes the phase-shift
\(C \rightarrow\) shifts the phase
\(D \rightarrow\) shifts the centre line.

Q.4. What are the five key points of a graph?
Ans: There are five key points that are required to graph a trigonometric function. That includes \(3x-\) intercepts, the maximum point, and the minimum point.

Q.5. How do you graph inverse trigonometric functions?
Ans: Steps to graph inverse trigonometric functions:
Step 1: Recall and draw the graph of the respective trigonometric functions.
Step 2: Identify the portion of the graph that you want to invert.
Step 3: Draw the line \(y=x\). This is the line of reflection.
Step 4: Reflect the graph of the trigonometric function across the line \(y=x\).
Step 5: The resulting graph is of the corresponding arc (inverse) function.

We hope this detailed article on the Graphs of Trigonometric Functions will make you familiar with the topic. If you have any inquiries, feel to post them in the comment box. Stay tuned to embibe.com for more information.