**$(x,y)=(-\sqrt 3,a)$****$(x,y)=(b,-\dfrac{\pi}{6})$****$(x,y)=(c,\dfrac{\pi}{4})$**

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) \[\dfrac{\pi}{3},1,-\dfrac{1}{\sqrt 2}\]**