- (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 $60^ {\circ}$.