4.07 Extension: Relating the tangent to the slope of a line
A special relationship exists between the tangent ratio and the slope of a line.
Exploration
Pictured here is a straight line on the $xy$xy-plane.
If we wanted to find the slope, we could look at $\frac{\text{rise }}{\text{run }}$rise run , which in this line is $\frac{AB}{CB}$ABCB.
Look at angle $\theta$θ, what is $\tan\theta$tanθ?
$\tan\theta=\frac{AB}{CB}$tanθ=ABCB. This is the value of the slope we figured out just before!
This means that $\tan\theta$tanθ, or specifically the tangent of the angle the line makes with the $x$x-axis in the positive direction, is equal to the slope.
The angle $\theta$θ is called the angle of inclination.
For a line with slope $m$m and angle of inclination $\theta$θ made with the positive $x$x-axis,
$\tan\theta=m$tanθ=m
Finding the angle of inclination from the slope
Worked example
Question 1
Evaluate: Suppose we know that a straight line has a slope of $3$3. What does this tell us about the line?
Think: Well we know that it is increasing, (positive slope), so has this shape. 
We also know that it is quite a bit steeper than the line $y=x$y=x, which has slope $1$1. We can also find the angle of inclination from this value. Remember that the value of the slope, in this case $3$3, is a measure of the $\frac{rise}{run}$riserun, so in this case the rise is $3$3, and the run is $1$1 as $3=\frac{3}{1}$3=31.
Do: We can draw a diagram of a triangle like this:
see how the rise and run form the vertical and horizontal components of our triangle. From this, we can calculate the angle of inclination, which is the angle marked $\theta$θ on this diagram.
| $\tan\left(\theta\right)$tan(θ) | $=$= | $\frac{opposite}{adjacent}$oppositeadjacent |
| $\tan\left(\theta\right)$tan(θ) | $=$= | $\frac{3}{1}$31 |
| $\theta$θ | $=$= | $\tan^{-1}\left(3\right)$tan−1(3) |
| $\theta$θ | $=$= | $71.565^{\circ}$71.565∘ |
Reflect: So the angle of inclination is $71.565^{\circ}$71.565∘
Finding the slope from the angle of inclination
Worked example
Question 2
Evaluate: Suppose we are told that the angle of inclination is $45^{\circ}$45∘. Use this information to determine the slope of the line.
Think: In this case, we start with the right triangle, the angle is given ($45^{\circ}$45∘) and we will add the adjacent side length of $1$1 unit. This is the horizontal run.
Do: Finding the vertical distance (the opposite side) is a process of trigonometry.
| $\tan\theta$tanθ | $=$= | $\frac{opposite}{adjacent}$oppositeadjacent | |
| $\tan\left(45^{\circ}\right)$tan(45∘) | $=$= | $\frac{opposite}{1}$opposite1 | (2) |
| $\tan\left(45^{\circ}\right)$tan(45∘) | $=$= | $opposite$opposite | |
| $opposite$opposite | $=$= | $1$1 |
Reflect: See on step (2), how by making the adjacent length $1$1 unit we were able to remove the fractional piece more easily?
Practice questions
Question 3
Two points $A$A$\left(1,-2\right)$(1,−2) and $B$B$\left(9,30\right)$(9,30) lie on a line that makes an angle of $\theta$θ degrees with the positive $x$x-axis.
Determine the slope $m$m of the line.
Find $\theta$θ in degrees to two decimal places.
Question 4
A line has slope measuring $-\frac{1}{\sqrt{3}}$−1√3, and makes an angle $\theta$θ, counterclockwise from the positive $x$x-axis to the line.
Solve for $\theta$θ in degrees.
Enter each line as an equation.