- $(x,y)=(-\sqrt 3,a)$
- $(x,y)=(b,-\dfrac{\pi}{6})$
- $(x,y)=(c,\dfrac{\pi}{4})$
The question aims to determine the missing coordinates of the points on the graph of the function y= arctan x.
A pair of numbers that shows the exact position of a point in a cartesian plane using horizontal and vertical lines called coordinates. It is usually represented by (x, y) the value of x and the y value of the point on the graph. Each topic or paired order contains two links. The first is x coordinate or abscissa, and the second is y axis or ordinate. Point link values can be any real positive or negative number.
Expert Answer
Part (a): For $(x,y)=(-\sqrt 3,a)$
The missing coordinate of the point on the graph pf the function $y=\arctan x$ is calculated as:
\[y=\arctan (-\sqrt 3)(-\sqrt 3,y)\]
\[y=-\dfrac{\pi}{3}\]
The output for the missing variable $a$ for the function $y=\arctan x$ is $(x,y)=(-\sqrt 3,-\dfrac{\pi}{3})$.
Part(b): For $(x,y)=(b,-\dfrac{\pi}{6})$
The missing $x-axis$ which is represented by the variable $b$ is calculated by using following procedure.
\[-\dfrac{\pi}{6}=\arctan (x)(x,-\dfrac{\pi}{6})\]
\[\tan(-\dfrac{\pi}{6})=x\]
\[x=-\dfrac{\sqrt 3}{3}\]
The output of the variable $b$ for the function $y=\arctan x$ is $(x,y)=(-\dfrac{\pi}{6},-\dfrac{\sqrt 3}{3})$.
Part(c): For $(x,y)=(c,\dfrac{\pi}{4})$
The missing value of the variable $c$ which is the value of the $x-axis$ is calculated by using the following method.
\[\tan\dfrac{\pi}{4}=x\]
\[x=1\]
The output of the variable $c$ for the function $y=\arctan x$ is $(x,y)=(1,\dfrac{\pi}{4})$.
The output is (from left to right) \[-\dfrac{\pi}{3},-\dfrac{\sqrt 3}{3},1\]
Numerical Result
The missing coordinates of the point for the graph of the function $y=\arctan x$ are calculated as:
Part (a)
$ (x,y)=(-\sqrt 3,a)$
The missing coordinate value is $-\dfrac{\pi}{3}$.
Part(b)
-$(x,y)=(b,-\dfrac{\pi}{6})$
The missing coordinate value is $-\dfrac{\sqrt 3}{3}$.
Part(c)
-$(x,y)=(c,\dfrac{\pi}{4})$
The missing coordinate value is $1$.
$-\dfrac{\pi}{3},-\dfrac{\sqrt 3}{3},1$
Example
Find the missing coordinates of the points on the graph of the functions: $y=cos^{-1} x$.
-$(x,y)=(-\frac{1}{2},a)$
-$(x,y)=(b,\pi)$
-$(x,y)=(c,\dfrac{\pi}{4})$
Part (a): For $(x,y)=(-\sqrt 2,a)$
The missing coordinate of the point on the graph pf the function $y=\arctan x$ is calculated as:
\[y=\cos^{-1} (-\dfrac{1}{2})(-\dfrac{1}{2},y)\]
\[y=\dfrac{\pi}{3}\]
The output of the missing variable $a$ for the function $y=\arctan x$ is $(x,y)=(-\dfrac{1}{2},\dfrac{\pi}{3})$.
Part(b): For $(x,y)=(b,\pi)$
The missing value of the variable $b$ that represents the $x-axis$ is calculated by using following procedure.
\[-\pi=\cos (x)(x,\pi)\]
\[\cos(\pi)=x\]
\[x=1\]
The output of the variable $b$ for the function $y=\arctan x$ is $(x,y)=(-\sqrt 3,\pi)$.
\[\dfrac{\pi}{4}=\arctan(x)(x,\dfrac{\pi}{4})\]
Part(c): For $(x,y)=(c,\dfrac{\pi}{4})$
The missing value of the variable $c$ that represents $x-axis$ is calculated by using the following method.
\[\cos\dfrac{\pi}{4}=x\]
\[x=\dfrac{1}{\sqrt 2}\]
The output is (from left to right) \[\dfrac{\pi}{3},1,-\dfrac{1}{\sqrt 2}\]