Tuesday, June 26, 2007

Fundamental Trigonometric Identities

Trigonometric identities are specific equalities that express one trigonometric function in terms of other trig functions. These fundamental trigonometric identities (also sometimes called basic trig identities) are fairly straightforward, but they take some work to derive them. If you are comfortable with simple derivations, you shouldn't have any problems though. Personally, I find it easier to remember the basic set of trig identities, and derive the more complex ones from those, rather than trying to memorize all of them... although some people are more comfortable just to memorize them.

These fundamental trigonometric identities are traditionally visualized as a right angle triangle inscribed within a circle of radius r, with sides formed by length x, height y, and the radius as the hypotenuse, r:

The basic trig definitions and relationships can easily be observed.  I have covered these trigonometric functions in other blog posts, so I won't go into them any further here, other than to just remind you of the concept of SOHCAHTOA.  Refer to these specific posts for more information about sine, cosine, and tangent.

Sin(θ) = y/r..... opposite  /hypotenuse
Cos(θ) = x/r..... adjacent / hypotenuse
Tan(θ) = y/x..... opposite / adjacent

For the first identity that I will show you how to derive, we need to make use of the Theorem of Pythagoras (I would also here like to refer you to my post about the Theorem of Pythagoras, which itself is derived from the cosine law, for more information on this trigonometric math concept.)  If we now apply the Theorem of Pythagoras to this right angle triangle we have inscribed within a circle, we can see:

r2 = x2 + y2
Dividing everything by r2 gives:
1 = (x2)/(r2) + (y2)/(r2)
1 = (x/r)2 + (y/r)2

And then, substituting in the basic trig identities listed above, we get:

1 = [Cos(θ)]2 + [Sin(θ)]2

And that is the first of the 8 fundamental trigonometric identities. Nothing to it. In fact, this is an important mathematical expression.  As such, it has a special name or designation given to it to indicate that it is important.  It's proper name is the Pythagorean Trigonometric Identity. I'll rewrite it below in proper notation to clean it up a bit.  Notice how you can write the 2 (for the squares) in two different ways.  Either as above, like [Cos(θ)]2, where you put the trig function in brackets to square it (note that you don't square the angle!), or more commonly as below, with the 2 placed next to the trig function to indicate that it is being squared.

Similarly, we can derive another basic relationship from a standard trigonometric identity.  For this one, it starts with:
Tan(θ) = y/x

Now to substitute in some more expressions.  This one is a little more complicated than how we arrived at the Pythagorean Identity above, but follow along and you should hopefully be able to see what I do.  The next step then, is to substitute in the basic Sine and Cosine definitions from the list at the top of this page (isolated for x and y, respectively) to give:

Tan(θ) = (r x Sin(θ)) / (r x Cos(θ))
Tan(θ) = Sin(θ) / Cos(θ)

And that's it again. Like the Pythagorean Trigonometric Identity I described above, this one also has a specific name.  This time, this one is called the Ratio Identity:

Those are now two of the simplest basic trig identities from which most of the others can be derived.  As I mentioned at the top, I always find that it is easier to memorize HOW to derive these fundamental trigonometric identities, rather than memorizing them explicitly and then possibly getting confused.  You WILL memorize them eventually (which always happens when you work with them so much!), but when you are first trying to understand them, I believe that it is always better to understand what is going on instead of just memorizing the final solutions.  This approach is a very wise strategy to apply throughout your mathematics training!

*Update 03-April-2012:  I have recently come back to this topic of basic trig identities, and have started a completely new post called the Trig Identities Cheat Sheet.  Please visit that site for a more thorough explanation of some of these derivations and fundamental identities of trigonometric functions.  Also, please click the +1 button below if you found this helpful!  Thanks.  


  1. Thanks so much for this blog! I am in grade 12 advanced functions and I still don't know how to do trig identities and we are learning them in the next 2 days. Since I didn't understand the basics from grade 11, I am trying to catch up on what I did not learn. Thanks again!

  2. Have a question - am learning these identities, and for my entire algebra training, I've been told you can't do to one side of an equals sign what you don't do to the other. i.e. if you do it on one side, you have to do it on the other. Yet, my math book is teaching me that it's okay (when proving an identity) to multiply the LHS by (say) a common denominator (numerator and denominator being multiplied) while not doing that same move on the RHS. This allows you to get a tidier problem, and now you can cancel, square, etc and come up with the "proof" of the identity... how does this work, since the RHS doesn't get the same treatment???

  3. Hi Denise,

    Sorry for the late reply.

    You have been taught correctly that what you do to one side, you must do to the other side. Add to them, multiply them, square them, whatever. Same treatment for each side keeps the expression equal.

    Your math book is correct, though at first glace it doesn't seem right. It is saying to multiply the numerator and denominator by the same factor, and this keeps everything equal. You can put this into familiar terms in one of two ways: (probably the key idea is #2)

    1) if you have x = a / b, and we say multiply top and bottom by 2, we get x = 2a / 2b. If you do regular arithmetic rearrangment, and multiply both sides by the denominator (2), you can see that you have essentially done the same to both the LHS and the RHS (that is, multiplied them both by 2).

    2) to multiply the numerator and denominator by the same factor is essentially multiplying the whole term by 1, because x / x equals 1, regardless of what it is. And so if you are multiplying the top and bottom by the same number, and essentially the whole expression by 1, you really aren't changing anything except to make it easier to work with.

    I hope this helps you with your confusion. One important distinction I should make as well is that when talking about RHS and LHS, you can do anything to both sides. Add to, subtract, multiply, square, etc. However, for the concept you are talking about, working with the numerator and denominator, you can only multiply or divide or square (or some form of these function)... you cannot add 3 to the top and 3 to the bottom, because that changes things. (eg. y = x/z = 2x/2z, but not (x+3 / z+3).


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