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

The key idea is that slope is the tangent of the angle the line makes with the positive x-axis, not the y-axis -- and the direction of measurement decides whether you add or subtract $90^\circ$.

Step 1: The line makes a $30^\circ$ angle with the positive y-axis, measured anticlockwise. Since the positive y-axis is already $90^\circ$ anticlockwise from the positive x-axis, rotating a further $30^\circ$ anticlockwise (away from the x-axis, not back toward it) carries the line to $90^\circ + 30^\circ = 120^\circ$ fro …

The key idea: slope is always defined relative to the x-axis. When a problem gives an angle with the y-axis, you must convert it to the angle with the x-axis before taking the tangent -- and the direction of measurement (clockwise vs anticlockwise) decides which way that conversion goes.

Why this works

The slope $m$ of a non-vertical line is defined as $m = \tan \theta$, where $\theta$ is the angle the line makes with the positive direction of the x-axis, measured anticlockwise, $0^\circ \le \theta \le 180^\circ$. This is the standard convention used throughout this chapter.

The positive x-axis and positive y-axis are $90^\circ$ apart (the y-axis lies $90^\circ$ anticlockwise from the x-axis). So if a line's angle with the y-axis is known, its angle with the x-axis can be found by adding or subtracting $90^\circ$ -- which operation applies depends on which direction the given angle is measured.

Step-by-step solution

1. Understand the given angle.

   The line makes an angle of $30^\circ$ with the positive direction of the y-axis, measured anticlockwise. "Measured anticlockwise" means: starting from the positive y-axis, rotate anticlockwise by $30^\circ$ to reach the line.

2. Find the angle with the x-axis.

   The positive y-axis itself is $90^\circ$ anticlockwise from the positive x-axis. Rotating a further $30^\circ$ anticlockwise from the y-axis carries the line past the y-axis into the second quadrant, so its angle with the positive x-axis is …