Question

How do you graph $y=\arccos \left( \dfrac{x}{3} \right)$?

Answer 1

In order to find the graph of the given equation in the question that is $y=\arccos \left( \dfrac{x}{3} \right)$, find the points that satisfy this equation and then plot them in the graph.

Complete step by step solution:
Given equation in the question is as follows
$y=\arccos \left( \dfrac{x}{3} \right)$
Consider the above equation as the function$f\left( x \right)=\arccos \left( \dfrac{x}{3} \right)$
To find the points that satisfy this equation first consider the point at $x=-3$
Replace the variable $x$ with $-3$ in the expression, we get:
$\Rightarrow f\left( -3 \right)=\arccos \left( \dfrac{-3}{3} \right)=\pi$
$\Rightarrow f\left( -3 \right)=\pi$
Now consider the point at $x=\dfrac{-3}{2}$
Replace the variable $x$ with $\dfrac{-3}{2}$in the expression, we get:
$\Rightarrow f\left( \dfrac{-3}{2} \right)=\arccos \left( \dfrac{-\dfrac{3}{2}}{3} \right)$
Simplify it further and multiply the numerator by the reciprocal of the denominator.
$\Rightarrow f\left( \dfrac{-3}{2} \right)=\arccos \left( -\dfrac{3}{2}\cdot \dfrac{1}{3} \right)$
Cancel the common factor of $3$ to do this follow the following steps:
Move the leading negative in $\dfrac{-3}{2}$into the numerator.
$\Rightarrow f\left( \dfrac{-3}{2} \right)=\arccos \left( -\dfrac{3}{2}\cdot \dfrac{1}{3} \right)$
Factor $3$ out of $-3$.
$\Rightarrow f\left( -\dfrac{3}{2} \right)=\arccos \left( \dfrac{3\left( -1 \right)}{2}\cdot \dfrac{1}{3} \right)$
Cancel the common factor.
$\Rightarrow f\left( -\dfrac{3}{2} \right)=\arccos \left( \dfrac{{3}\cdot -1}{2}\cdot \dfrac{1}{{{3}}} \right)$
Rewrite the expression, we get:
$\Rightarrow f\left( -\dfrac{3}{2} \right)=\arccos \left( \dfrac{-1}{2} \right)$
Move the negative in front of the fraction.
$\Rightarrow f\left( -\dfrac{3}{2} \right)=\arccos \left( -\dfrac{1}{2} \right)$
The exact value of $\arccos \left( -\dfrac{1}{2} \right)$ is $\dfrac{2\pi }{3}$.
$\Rightarrow f\left( -\dfrac{3}{2} \right)=\dfrac{2\pi }{3}$
After this consider the point at $x=0$
Replace the variable $x$ with $0$ in the expression, we get:
$\Rightarrow f\left( 0 \right)=\arccos \left( \dfrac{0}{3} \right)$
Simplify it further and divide $0$ by $3$, we get:
$\Rightarrow f\left( 0 \right)=\arccos \left( 0 \right)$
The exact value of $\arccos \left( 0 \right)$ is $\dfrac{\pi }{2}$.
$\Rightarrow f\left( 0 \right)=\dfrac{\pi }{2}$
Now find for the point at $x=\dfrac{3}{2}$.
Replace the variable $x$ with $\dfrac{3}{2}$ in the expression, we get:
$\Rightarrow f\left( \dfrac{3}{2} \right)=\arccos \left( \dfrac{\dfrac{3}{2}}{3} \right)$
Simplify it further and multiply the numerator by the reciprocal of the denominator.
$\Rightarrow f\left( \dfrac{3}{2} \right)=\arccos \left( \dfrac{3}{2}\cdot \dfrac{1}{3} \right)$
Cancel the common factor of $3$, we get
$\Rightarrow f\left( \dfrac{3}{2} \right)=\arccos \left( \dfrac{1}{2} \right)$
The exact value of $\arccos \left( \dfrac{1}{2} \right)$is $\dfrac{\pi }{3}$.
$\Rightarrow f\left( \dfrac{3}{2} \right)=\dfrac{\pi }{3}$
Now at last find the point at $x=3$
Replace the variable $x$with $3$ in the expression, we get:
$\Rightarrow f\left( 3 \right)=\arccos \left( \dfrac{3}{3} \right)$
Simplify it further and divide $3$ by $3$, we get:
$\Rightarrow f\left( 3 \right)=\arccos \left( 1 \right)$
The exact value of $\arccos \left( 1 \right)$is $0$.
$\Rightarrow f\left( 3 \right)=0$
Now list all the points in a table and this trigonometric function can be graphed by plotting these points.

x & f\left( x \right) \\\ -3 & \pi \\\ -\dfrac{3}{2} & \dfrac{2\pi }{3} \\\ 0 & \dfrac{\pi }{2} \\\ \dfrac{3}{2} & \dfrac{\pi }{3} \\\ 3 & 0 \\\ \end{matrix}$$ The graph of the function $$f\left( x \right)=\arccos \left( \dfrac{x}{3} \right)$$ is as follows:  **Note:** Students generally make calculation mistakes while calculating the points that satisfy the graph. It's important to remember the trigonometric values and recheck the calculations once done.