Inclination of a line

The angle of inclination $\theta$ of a line is the angle between the line and the positive $x$-axis, measured anticlockwise, with $0°\le \theta <180°$.

$m=\tan \theta$

• Positive gradient: $\theta$ is acute. $m=1$ gives $\theta =45°$.
• Negative gradient: $\theta$ is obtuse. The calculator gives a negative angle, so add 180°.

Example. $2x+3y=6$ gives $y=-\frac{2}{3}x+2$, so $m=-\frac{2}{3}$.

${\tan }^{-1}\left(-\frac{2}{3}\right)\approx -33,69°\Rightarrow \theta \approx 180°-33,69°=146,31°$

Horizontal lines have $\theta =0°$. Vertical lines have $\theta =90°$ and an undefined gradient.

Two lines and the $x$-axis form a triangle. Use the exterior angle of a triangle: the larger inclination equals the smaller inclination plus the angle between the lines.

$\text{angle between the lines}={\theta }_2-{\theta }_1$

Example. $y=2x+1$ has ${\theta }_1={\tan }^{-1}2\approx 63,43°$. $y=-x+4$ has ${\theta }_2=135°$. The angle between them is $135°-63,43°=71,57°$.

If the result is obtuse, the acute angle between the lines is $180°$ minus it.

Working backwards. A line through $P(-2;1)$ at 60° to the positive $x$-axis has $m=\tan 60°=\sqrt{3}$, so $y-1=\sqrt{3}(x+2)$.