Question

# Find the slope of the line, which makes an angle of $30^\circ$ with the positive direction of y-axis measured anticlockwise.

Hint: Need to visualize the given information on coordinate axes. Slope of a line is measured with the help of the angle measured from the line with respect to the positive x-axis.

Given that a line makes an angle of $30^\circ$ with the positive y-axis measured anti clockwise.

That means the corresponding figure will be like,

We need to find the $\theta$, which is measured from the line with respect to positive x-axis.

Then $\theta = 30^\circ + 90^\circ = 120^\circ$
Thus, slope of the given line is $\tan \theta = \tan 120^\circ$
$\Rightarrow \tan 180^\circ = \tan (180^\circ - 60^\circ ) = - \tan 60^\circ = - \sqrt 3$
$\therefore$ The slope of the given line is $- \sqrt 3$.

Note: To find the slope of a line, we find the inclination angle and apply a tangent to that angle to give the slope. Inclination angle is the angle measured with positive x-axis and the line. We need to know the basic trigonometric function values to solve these kinds of problems.
We used values: $\tan (180 - \theta ) = - \tan \theta$, $\tan 60^\circ = \sqrt 3$.