Question

Find the inclination of a line whose slope is:

(A). \[\sqrt{3}\]
(B). \[\dfrac{1}{\sqrt{3}}\]
(C). \[1\]

Answer 1

Hint: Use the fact that the slope of the line can also be represented as the tangent of the angle which the line makes with the positive x-axis when going anticlockwise from the x-axis. The value of m gives the slope of the line and then equate it to the tangent of the angle which the line makes with the positive x-axis when going anticlockwise from the x-axis as follows \[m=\tan \theta \] (Where \[\theta \] is the angle that the line makes with the positive x-axis when going anticlockwise from the x-axis and m is the slope of the line which is inclined to the x-axis with the mentioned angle)

Complete step-by-step answer:

Now, in this question, we will simply put the value of the slope that is given in the question and then we will get the value of the inclination on taking or finding the tan inverse of that slope.

As mentioned in the question, we have to find the slope of the line which makes the given angle with the x-axis when going anticlockwise from the x-axis.
A. \[\sqrt{3}\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & m=\tan \theta \\
 & \sqrt{3}=\tan \left( \theta \right) \\
 & \theta ={{\tan }^{-1}}\sqrt{3} \\
 & \theta ={{60}^{\circ }} \\
\end{align}\]
Hence, the angle of inclination of the line which has the mentioned slope, with the x-axis, is \[{{60}^{\circ }}\] .
B. \[\dfrac{1}{\sqrt{3}}\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & m=\tan \theta \\
 & \dfrac{1}{\sqrt{3}}=\tan \left( \theta \right) \\
 & \theta ={{\tan }^{-1}}\dfrac{1}{\sqrt{3}} \\
 & \theta ={{30}^{\circ }} \\
\end{align}\]
Hence, the angle of inclination of the line which has the mentioned slope, with the x-axis, is \[{{30}^{\circ }}\]
C. \[1\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & m=\tan \theta \\
 & 1=\tan \left( \theta \right) \\
 & \theta ={{\tan }^{-1}}1 \\
 & \theta ={{45}^{\circ }} \\
\end{align}\]
Hence, the angle of inclination of the line which has the mentioned slope, with the x-axis, is \[{{45}^{\circ }}\] .


Note:
The students can make an error if they don’t know about the formulae that are given in the hint as without knowing them one can never get to the correct answer.
Also, in this question, it is important to be extra careful while doing the calculations as it might be possible that the students end up getting a wrong result due to some calculation mistakes.
Also, it is important to know the basic values and the basic properties of tangent function and inverse tangent function for solving this question as without knowing them one can never get to the correct answer.