Angles made by the line with the positive direction of X-axis are given. Find the slope of these lines. (1) 45° (2) 60° (3) 90°

Explanation:–

The slope of a line is defined as the ratio of the change in the vertical coordinate (y-coordinate) to the change in the horizontal coordinate (x-coordinate) between any two points on the line.

Given the angle that the line makes with the positive direction of the X-axis, we can use trigonometric functions to determine the slope of the line.

Let’s consider each angle given:

(1) 45°: A line that makes an angle of 45° with the positive direction of the X-axis forms an isosceles right triangle with the X-axis. The slope of the line is equal to the tangent of this angle.

Tangent of 45° is 1. Therefore, the slope of the line that makes an angle of 45° with the positive direction of the X-axis is 1.

(2) 60°: A line that makes an angle of 60° with the positive direction of the X-axis forms an equilateral triangle with the X-axis. The slope of the line is equal to the tangent of this angle.

Tangent of 60° is sqrt(3). Therefore, the slope of the line that makes an angle of 60° with the positive direction of the X-axis is sqrt(3).

(3) 90°: A line that makes an angle of 90° with the positive direction of the X-axis is a vertical line. The slope of a vertical line is undefined.

Therefore, the slope of the line that makes an angle of 45° with the positive direction of the X-axis is 1, the slope of the line that makes an angle of 60° with the positive direction of the X-axis is sqrt(3), and the slope of the line that makes an angle of 90° with the positive direction of the X-axis is undefined.