Question

Angle of line with +ve direction of x-axis is $\theta$. The line is rotated about some point in it in anticlockwise direction by angle ${45}^{{}^0}$and its slope become $$3,Find\theta .$$

Answer 1

Hint:We should have knowledge of slope of a straight line & some basic knowledge of trigonometry. Formula of slope of a straight line should be applied & solved to get the angle between that line & positive x axis.

Complete step-by-step answer:
 Given, the angle of line with the +ve direction of the x-axis is $\theta$.
We know , slope or gradient is defined as a number that describes both the direction & steepness of the line. The slope is represented by $\mathrm{m}=tan\theta$, where m is the slope of that line which forms $\theta$ angle with the x-axis .
When a line is rotated anticlockwise from +ve x axis, it forms an angle in the 1st quadrant.
As per question , let's suppose the line AB is rotated about point A in anticlockwise direction by angle ${45}^0$ and become AC.
Now, applying $\mathrm{m}=tan\theta$ by the definition of slope $forAC$
$\mathrm{Slope}\left(m\right)=tan\theta =3$
$\therefore tan\left(45+\theta \right)=3$
$\Rightarrow 45+\theta =71.565$ [ taking inverse of tan as we know if $$\tan \theta =x$$ then $\theta ={\tan }^{-1}x$ by using calculator, you can get value of inverse of tan for given value]
$\Rightarrow \theta =71.565-45$
$\Rightarrow 45+\theta =71.565$ [ solving for $\theta$ ]
$\Rightarrow \theta =26.565=26.57^0\left(approximately\right)$
Hence those lines make an angle of ${26.57}^{\circ }$ with the +ve x-axis.

Note:In this type of problem, the angle of the straight line varies when the line is rotated clockwise or anticlockwise in any direction.The slope is represented by $\mathrm{m}=tan\theta$, where m is the slope of that line which forms $\theta$ angle with the x-axis .