Drawing angles on a graph

Question

Is there an easy way to draw lines on a graph to be at certain angles using only rise/run? For example, to make a 45 degree angle with the x axis, the rise/run = 1.

Answer 1

If you have a line of slope $m = {\text{rise}}/{\text{run}}$, and $\theta$ is the angle it makes with the x-axis, then $\tan(\theta) = m$.

This is because if you draw a right triangle from the angle, the rise/run ratio is the same as the side opposite of the angle divided by the adjacent side. This is the tangent function in trigonometry.

So you can use $\tan^{-1}(m)$, the inverse tangent function (sometimes written as the "$\text{arctangent}$", or "$\arctan(m)$"), to find the measure of the angle $\theta$.

Most scientific calculators will have an inverse tangent function. Microsoft Excel also has an ATAN(number) function, though it gives the resulting angle in radians. You have to convert it to degrees by entering =+DEGREES(ATAN(1)) for example to get back 45.

Answer 2

The slope (rise/run) of a line which forms an angle $\theta$ with the positive x-axis is $\tan \theta$, which you can calculate using a scientific calculator or a website of your choice. Unfortunately, for most angles this is not a nice number.