Trigonometry—Graphing the Sine, Cosine and Tangent Functions
Graphing trigonometric functions
Trig graphs are easy once you get the hang of them. Once you learn the basic shapes, you shouldn't have much difficulty.
The main problems A-Level students have, in my experience, are:
- Remembering which is y = sin x and which is y = cos x. There's a trick to this I'll cover in a minute.
- Recalling the values of the asymptotes on the graph of y = tan x. Again, there are a couple of simple tips for making this easier.
Sine and cosine graphs
y = sin x and y = cos x look pretty similar; in fact the main difference is that the sine graph starts at (0,0) and the cosine at (0,1).
Top tip for the exam: To check you've drawn the right one, simply use your calculator to find sin 0 (which is 0) or cos 0 (which is 1) to make sure you're starting in the right place!
Both of these graphs repeat every 360 degrees, and the cosine graph is essentially a transformation of the sin graph - it's been translated along the x-axis by 90 degrees. Thinking about the fact that sin x = cos (90 - x) and cos x = sin (90 - x), it makes pretty good sense that they're 90 degrees out of phase.
The graph of y = tan x is an odd one - mainly down to the nature of the tangent function. Going back to SOH CAH TOA trig, with tan x being opposite / adjacent, you can see that:
Tan 0 = 0, as the opposite side would have zero length regardless of the length of the adjacent side.
Tan 90 is not possible, as we can't have a triangle with two right angles! As the angle approaches 90 degrees, our opposite side would approach inifinity.
This means that the graph of y = tan x crosses the x-axis at 0, and has an asymptote at 90. This graph repeats every 180 degrees, rather than every 360 (or should that be as well as every 360?)
Using tan x = sin x / cos x to help
If you can remember the graphs of the sine and cosine functions, you can use the identity above (that you need to learn anyway!) to make sure you get your asymptotes and x-intercepts in the right places when graphing the tangent function.
At x = 0 degrees, sin x = 0 and cos x = 1. Tan x must be 0 (0 / 1)
At x = 90 degrees, sin x = 1 and cos x = 0. Tan x has an asymptote (1 / 0)
At x = 180 degrees, sin x = 0 and cos x = 1. Tan x must be 0 (0 / 1)
At x = 270 degrees, sin x = 1 and cos x = 0. Tan x has an asymptote (1 / 0)
...and so on!