Question

Find the slope of the line which make the following angles with the positive direction off x-axis:-
A) \[\dfrac{-\pi }{4}\]
B) \[\dfrac{2\pi }{3}\]
C) \[\dfrac{3\pi }{4}\]
D) \[\dfrac{\pi }{3}\]

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 angle that is given in the question and then we will get the value of the slope on taking or finding the tangent of that angle.
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) \[\dfrac{-\pi }{4}\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & \Rightarrow m=\tan \theta \\
 & \Rightarrow m=\tan \left( \dfrac{-\pi }{4} \right) \\
 & \left[ \tan (-x)=-\tan x \right] \\
 & \Rightarrow m=-\tan \left( \dfrac{\pi }{4} \right) \\
 & \Rightarrow m=-1 \\
\end{align}\]
Hence, the slope of the line which makes the mentioned angle with the x-axis, is -1.
B) \[\dfrac{2\pi }{3}\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & \Rightarrow m=\tan \theta \\
 & \Rightarrow m=\tan \left( \dfrac{2\pi }{3} \right) \\
 & \Rightarrow m=\tan \left( \pi -\dfrac{\pi }{3} \right) \\
 & \left[ \tan \left( \pi -x \right)=-\tan x \right] \\
 & \Rightarrow m=-\tan \left( \dfrac{\pi }{3} \right) \\
 & \Rightarrow m=-\sqrt{3} \\
\end{align}\]
Hence, the slope of the line which makes the mentioned angle with the x-axis, is \[-\sqrt{3}\].
C) \[\dfrac{3\pi }{4}\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & \Rightarrow m=\tan \theta \\
 & \Rightarrow m=\tan \left( \dfrac{3\pi }{4} \right) \\
 & \Rightarrow m=\tan \left( \pi -\dfrac{\pi }{4} \right) \\
 & \left[ \tan \left( \pi -x \right)=-\tan x \right] \\
 & \Rightarrow m=-\tan \left( \dfrac{\pi }{4} \right) \\
 & \Rightarrow m=-1 \\
\end{align}\]
Hence, the slope of the line which makes the mentioned angle with the x-axis, is -1.
D) \[\dfrac{\pi }{3}\]
We know that the slope of the line can be calculated as follows
\[\begin{align}
  & \Rightarrow m=\tan \theta \\
 & \Rightarrow m=\tan \left( \dfrac{\pi }{3} \right) \\
 & \Rightarrow m=\sqrt{3} \\
\end{align}\]
Hence, the slope of the line which makes the mentioned angle with the x-axis, is \[\sqrt{3}\].
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, knowing these following important relations is also very essential which are
\[\begin{align}
  & \Rightarrow \tan (\pi -x)=-\tan (x) \\
 & \Rightarrow \tan (-x)=-\tan \left( x \right) \\
\end{align}\]
Also, it is important to correctly do the calculation part as it might be possible that the students commit a mistake.