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Trigonometry—Graphing the Sine, Cosine and Tangent Functions

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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:

For each question, choose the best answer for you.

  1. Which graph peaks at 0 and 360? (without looking!)
    • y = sin x
    • y = cos x
    • y = tan x
  2. Which is constrained to y values between -1 and 1?
    • y = sin x
    • y = cos x
    • y = tan x
  3. Which graph crosses the x axis at 90 and 270?
    • y = sin x
    • y = cos x
    • y = tan x
  4. Which crosses the x axis at 180 and 360?
    • y = sin x
    • y = cos x
    • y = tan x
  5. Which is symmetrical about x = 90?
    • y = sin x
    • y = cos x
    • y = tan x

Scoring

For each answer you selected, add up the indicated number of points for each of the possible results. Your final result is the possibility with the greatest number of points at the end.

  1. Which graph peaks at 0 and 360? (without looking!)
    • y = sin x
      • doing great!: -3
      • getting mixed up,: +1
      • getting confused,: 0
    • y = cos x
      • doing great!: +1
      • getting mixed up,: 0
      • getting confused,: 0
    • y = tan x
      • doing great!: -3
      • getting mixed up,: 0
      • getting confused,: +1
  2. Which is constrained to y values between -1 and 1?
    • y = sin x
      • doing great!: +1
      • getting mixed up,: 0
      • getting confused,: 0
    • y = cos x
      • doing great!: +1
      • getting mixed up,: 0
      • getting confused,: 0
    • y = tan x
      • doing great!: -3
      • getting mixed up,: 0
      • getting confused,: +1
  3. Which graph crosses the x axis at 90 and 270?
    • y = sin x
      • doing great!: -2
      • getting mixed up,: +1
      • getting confused,: 0
    • y = cos x
      • doing great!: +1
      • getting mixed up,: 0
      • getting confused,: 0
    • y = tan x
      • doing great!: -3
      • getting mixed up,: 0
      • getting confused,: +1
  4. Which crosses the x axis at 180 and 360?
    • y = sin x
      • doing great!: -2
      • getting mixed up,: 0
      • getting confused,: +1
    • y = cos x
      • doing great!: -2
      • getting mixed up,: 0
      • getting confused,: +1
    • y = tan x
      • doing great!: +1
      • getting mixed up,: 0
      • getting confused,: 0
  5. Which is symmetrical about x = 90?
    • y = sin x
      • doing great!: +1
      • getting mixed up,: 0
      • getting confused,: 0
    • y = cos x
      • doing great!: -3
      • getting mixed up,: +1
      • getting confused,: 0
    • y = tan x
      • doing great!: -3
      • getting mixed up,: 0
      • getting confused,: +1

This table shows the meaning of each possible result:

doing great!

You know your stuff, well done!

getting mixed up,

but don't stop trying! You're confusing your sine and cosine graphs, would it help to sketch them a few times?

getting confused,

but don't worry! It's not an easy topic at first. Practice sketching the graphs and marking on the important values at 0, 90, 180, 270 and 360.

Comments

Small on November 11, 2018:

Cheers

TEOPOLINA on March 17, 2014:

I WANNA SAY THANKS A lot TO THE CREATER OT THIS PAGE . IT REALLY HELPED ME OUT WITH MY HOMEWORK

Mr Homer (author) from Yorkshire, England on January 18, 2011:

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!

Natalie on December 09, 2010:

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

Ben Evans on May 30, 2010:

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

Cheers,

Ben

Jen on April 06, 2010:

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

Mr Homer (author) from Yorkshire, England on March 24, 2010:

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

Glad you found it useful,

Tom

catalystsnstars from Land of Nod on March 23, 2010:

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