Gradient as tan of the inclination
If a line makes angle θ with the positive x-axis, its gradient is tan θ.
Key formulas
Meaning
θ = 0° → horizontal → m=0. θ=45° → m=1. θ=90° is vertical and tan θ undefined.
Example
A road rises at 30° to the horizontal → gradient tan30°=1/√3.
Two lines
The angle α between lines with gradients m₁, m₂ satisfies tan α = |(m₁−m₂)/(1+m₁m₂)| when defined.
Worked example
Worked example
Problem. A line makes an angle of 60° with the positive x-axis. Find its gradient.
Solution. m = tan 60° = √3.
Negative
θ in the second quadrant sense gives negative tan for downward slopes in the usual diagram.
Review quiz
Inclination 30°. Gradient is:
Practice questions
- Find m when θ=60°.
- A line has m=√3. Find its inclination acute angle.
Study notes
Degrees?
Yes for these geometry angles unless stated otherwise.