286. Angle of the Line · How steep is your line?

Try to break it

Drag P around O. The line through OP rotates, and the inclination θ (measured from the positive x-axis) cycles through [0°, 180°). Past the vertical (90°) the line tilts the other way and θ keeps growing toward 180°, where it wraps back to 0°. Horizontal line: θ = 0°. Vertical line: θ = 90°.

How you build it

Draw lines at different inclinations to the x-axis.

• Set the angle dial to 30°. Click the origin O, then click a point to the right along the x-axis. The ray rises 30° above the x-axis — a 30° inclination.
• Set the dial to 90°. Click O, then a point to the right on the x-axis. The ray points straight up — a vertical line has inclination 90°.
• Set the dial to 135°. Click O, then a point to the right on the x-axis. The ray tilts up to the left — inclinations run 0° up to (but not including) 180°.

The proof, step by step

Prove that the inclination is the angle the line makes with the positive x-axis.

• The inclination θ is measured from the positive direction of the x-axis.
• The measurement is always taken in the anti-clockwise direction.
• For any line parallel to the x-axis, θ = 0°.
• For any line parallel to the y-axis, θ = 90°.

Worked example

A straight line makes an angle of 45° with the positive direction of the x-axis. What is the inclination of the line?

By definition, the inclination of a line is the angle it makes with the positive direction of the x-axis measured anti-clockwise. Since the line makes 45° with the positive x-axis, its inclination is exactly 45°.

• 45° — correct
• 135°
• 90°
• 0°