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Verifying Trigonometric Identities with Unit Circles

Instructor: Carmen Andert

Carmen has two master's degrees in mathematics has has taught mathematics classes at the college level for the past 9 years.

In this lesson, we will review the unit circle and the corresponding definitions of the trigonometric functions. We will then verify several trigonometric identities using the unit circle.

Verifying Trigonometric Identities

Mathematicians use trigonometric identities all the time, and trigonometry students are forced to memorize them. But how do we know they are true? Just because your math teach says something does not necessarily make it true. If she says that all dinosaurs were purple, should you believe her? (Probably not!) What if she says that sin2 (t) + cos2 (t) = 1? should you believe her? (Yes, you should believe that one!)

Trigonometric Identities are equations involving one or more trigonometric functions. To verify or prove an identity, we must use our knowledge of the unit circle and the definitions of the trigonometric functions to show that the equation is true for all values of the variable. The variable may be any symbol, most commonly θ, φ, x, or t. In this lesson, we will stick with t as the variable.

The Unit Circle and the Trigonometric Functions

The Unit Circle is the circle centered at the origin with radius 1. The equation for the unit circle is x2 + y2 = 1. In our lesson, t represents an angle measured counterclockwise from the positive x-axis. For a given value of t, we will let (x, y) be the point where the ray at angle t intersects the unit circle, as shown in the diagram below.


The Unit Circle
unit circle


With this setup, we define the six trigonometric functions as follows:

  • sin(t) = y
  • cos(t) = x
  • tan(t) = y/x
  • sec(t) = 1/x
  • csc(t) = 1/y
  • cot(t) = x/y

Verifying Trigonometric Identities

To verify any trigonometric identity using the unit circle, you follow the same set of guidelines:

  1. Choose a side to start with. (Usually the more complicated side.) Write it down.
  2. Convert all Trigonometric functions to expressions involving x and y using the definitions above. It may also be helpful to convert the other side in the same way to see where you are headed, but just write this down on scratch paper for now.
  3. Use algebra and the equation x2 + y2 = 1 to change what you currently have to what you need for the other side.
  4. Using the definitions, change all the x's and y's back to trigonometric functions to match the other side.

Essentially, you start with one side of the identity, do some work, and end up with the other side.

Example

Remember when your teacher told you that sin2 (t) + cos2 (t) = 1? This is how we can verify that she wasn't lying:

sin2 (t) + cos2 (t) =1

Since the left side is more complicated, we will start with that.

Mathematical Step: Reason
sin2 (t)+ cos2 (t) = y2+x2 Substitute y for sin(t) and x for cos(t) based on above table
y2 + x2 = x2 + y2 Commutative property of addition--you can change the order
x2 + y2= 1 Equation of the unit circle

Since we started with one side and ended with the other, we are done!

Example

One more easy one: Let's prove that cot(t) = cos(t)/sin(t).

Let's start with the right side this time since it is a little bit more complicated.

Mathematical Step: Reason
cos(t)/sin(t) = x/y By definition of cos(t) and sin(t)
x/y = cot(t) Definition of cot(t)

Done! That one was even easier that the first one. Time to make it harder...

Example

Prove that sec(t) + tan(t) = cos(t)/( 1 - sin(t) ).

This one is a bit more complicated. Let's start with the right side, but on some scratch paper, notice that the left side is:

sec(t) + tan(t) = 1/x + y/x

Now we know what we are aiming for as we work.

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