Question

Given that slope is $- 1$. How to find the angle?

Answer 1

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.