Question

Given that slope is \[ - 1\]. How to find the angle?

Answer 1

Hint: First, we will use the formula is \[m = \tan \theta \], where \[\theta \] is the angle line makes with the \[x\]–axis and \[m\] is the slope of the line. Then we will simplify them using \[{\tan ^{ - 1}}\tan m = m\] to find the required angle.

Complete step-by-step answer:
We are given that the slope is \[ - 1\].
We know that the formula is \[m = \tan \theta \], where \[\theta \] is the angle line made with the \[x\]–axis and \[m\] is the slope of the line.
Substituting the value of \[m\] in the above formula of angle, we get
\[ \Rightarrow - 1 = \tan \theta \]
Applying the \[{\tan ^{ - 1}}\] in the above equation, we get
\[ \Rightarrow {\tan ^{ - 1}}\left( { - 1} \right) = {\tan ^{ - 1}}\tan m\]
Using the trigonometric value, \[{\tan ^{ - 1}}\tan m = m\] in the above equation, we get
\[
   \Rightarrow {\tan ^{ - 1}}\left( { - 1} \right) = m \\
   \Rightarrow m = {\tan ^{ - 1}}\left( { - 1} \right) \\
 \]
Using the value of \[{\tan ^{ - 1}}\left( { - 1} \right) = 45^\circ \] in the above equation, we get
\[ \Rightarrow m = 45^\circ \]
Thus, the required angle is \[45^\circ \].


Note: In this question, we can also solve it by taking the angle made by \[y = mx\] with positive direction of \[x\]–axis is \[{\tan ^{ - 1}}m\] and the angle made by line by \[y = nx\] is \[{\tan ^{ - 1}}n\].
Now, take \[{\tan ^{ - 1}}m = 2{\tan ^{ - 1}}n\].
\[
   \Rightarrow {\tan ^{ - 1}}m = {\tan ^{ - 1}}\dfrac{{2n}}{{1 - {n^2}}} \\
   \Rightarrow m = \dfrac{{2n}}{{1 - {n^2}}} \\
 \]
Substituting this value of \[m\] in the equation \[m = 4n\] and continue in the similar manner as given in the solution.