**(1, $\sqrt{3}$)****(2, 4)****(-$\sqrt{3}$, 3)**

The question aims to find the **point** on the **cartesian plane** of a given **angle** on the **terminal side.**

The question is based on the concept of **trigonometric ratios. Trigonometry** deals with a **right-angle triangle,** its **sides,** and angle with its **base.**

## Expert Answer

The given information about this problem is given as:

\[ \theta = -210^ {\circ} \]

Different **points** of the **terminal side** are given and we need to find the **correct** one. We can use the $\tan$ identity to check the value of the given **angle** and match it with the given points.

The **trigonometric identity** is given as:

\[ \tan \theta = \dfrac{ y }{ x } \]

\[ \tan (-210^ {\circ}) = \dfrac{ y }{ x } \]

\[ \dfrac{ y }{ x } = – \dfrac{ \sqrt {3} }{ 3 } \]

**a)** (1, $\sqrt{3}$)

Here, we replace the **values** of **x** and **y** and simplify them to see if it equals the desired **result.**

\[ \dfrac{ y }{ x } = \dfrac{ 1 }{ \sqrt {3} } \]

This point is **not** on the **terminal side** of $-210^ {\circ}$.

**b)** (2, 4)

\[ \dfrac{ y }{ x } = \dfrac{ 4 }{ 2 } \]

\[ \dfrac{ y }{ x } = 2 \]

This point is **not** on the **terminal side** of $-210^ {\circ}$.

**c) **($\sqrt{3}$, 3)

\[ \dfrac{ y }{ x } = \dfrac{ \sqrt {3} }{ 3 } \]

This point **lies** on the **terminal side** of $-210^ {\circ}$.

## Numerical Result

The **point** (-$\sqrt{3}$, 3) lies on the **terminal side** of $-210^ {\circ}$.

## Example

Choose the **point** on the **terminal side** of $60^ {\circ}$.

– (1, $\sqrt{3}$)

– ($\sqrt {3}$, 1)

– (1, 2)

Calculating the **value** of the **tangent** of $60^ {\circ}$, which is given as:

\[ \tan (60^ {\circ} = \dfrac{ y }{ x } \]

\[ \dfrac{ y }{ x } = \sqrt {3} \]

**a)** (1, $\sqrt{3}$)

\[ \dfrac{ y }{ x } = \dfrac{ 1 }{ \sqrt {3} } \]

This point is **not** on the **terminal side** of $60^ {\circ}$.

**b)** ($\sqrt {3}$, 1)

\[ \dfrac{ y }{ x } = \dfrac{ \sqrt {3} }{ 1 } \]

\[ \dfrac{ y }{ x } = \sqrt {3} \]

This **point lies** on the **terminal side** of $60^ {\circ}$.

**c)** (1, 2)

\[ \dfrac{ y }{ x } = \dfrac{ 1 }{ 2 } \]

This point is **not** on the **terminal side** of $60^ {\circ}$.