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.

sine, cosine and tangent graphs - remember the key points: 0, 90, 180, 270, 360 (click to enlarge)
sine, cosine and tangent graphs - remember the key points: 0, 90, 180, 270, 360 (click to enlarge)

Tangent graphs

 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!

Take the trig graphs quiz:


catalystsnstars profile image

catalystsnstars 6 years ago from Land of Nod

You remind me of my brother, thanks for the short tutor session. Thumbs up on going through with it.

Mr Homer profile image

Mr Homer 6 years ago from Yorkshire, England Author

Thanks, I think! At some point this hub should link to other trigonometry topics, so keep coming back.

Glad you found it useful,


Jen 6 years ago

thanks for this ! I'm having some trouble in my Algebra 2/Trig H class.

Ben Evans profile image

Ben Evans 6 years ago

Very nice. I wish I could figure out how to publish graphics. This will be helpful for those wanting to learn trigonometry.



Natalie 5 years ago

Do you know of a real world application of trig graphing? I am doing a precal project and i am trying to find a job in the aviation realm besides the pilot that uses this concept. i was thinking a tool and die maker, but i cannot find any concrete examples of them using this concept...any ideas? thanks

Mr Homer profile image

Mr Homer 5 years ago from Yorkshire, England Author

Thanks for the comments folks.

Natalie- I know that basic light/sound waves follow a sin-type curve, and there are also plenty of applications to do with circular motion... but as for jobs for your project, I can't really think of anything. Sorry!

TEOPOLINA 2 years ago


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