Proving Trig Identites

You have in your possession an amazing tool called your graphing calculator.  While the TI 84 can't do algebra for you (but there are "CAS" algebra systems on calculators....just not allowed on your standardized tests), the TI 84 can help you simplify or verify identities using the graphing application of the calculator.

Let's start with those exercises where you need to "simplify" a trig statement.

The idea: if you are looking at one of those trig statements that looks crazy busy and are not sure where to go, sometimes it's easier to work through finding the answer (that is, writing the *solution*) if you know the answer. But darned it if the teachers want you to "show your work" and not just guess, so you still need to do that algebra to the trig functions.  So how can you find the answer so you can be helped to find the solution?

Let's say you are asked to simplify: (sinx+cosx)^2-2secxcscx and you don't know where to start.
Type the expression into Y1, then graph.   You'll want to be conscious of what mode you want to consider; but you'll find that the graph of this expression is simply a horizontal line at y = 1.  That information suggests to you that the expression simplifies to 1.

Now how about proving trig identities?

To review the definition of identity: An equation that is true for ALL values of the variables; not just one or two, but for all values of "x."

Perhaps you've been trying to prove an identity and have found yourself in algebra hell for a while and you're not sure if the equation is an identity after all.  How might your calculator help determine if your teacher or the text made a mistake?

This is your question to answer for your blog.  Start with the following link:
http://mathbits.com/MathBits/TISection/Trig/trigidentity.htm

In your blog, find an equation that is NOT an identity. It could be an equation with one or more solutions or it could be an equation with no solutions.   Show how your calculator would disprove the equation as an identity. Be creative. Have fun.

Then find your own identity to use. Use the basic identities or the Pythagorean identities to help you find one.  Again, be creative. Show how you would use your calculator to strongly suggest that the statement IS an identity.   Notice my change in wording...  Can't you use your calculator's graphing (or tables) tools to prove that an equation is an identity?  You'll need to address that also.

In sum:
a. Find an equation that is not an identity.  Use your calculator to demonstrate that it's not an identity. Be creative. Take screen shots, pictures, images, etc.
b. Find an identity.  Use your calculator to show what an identity looks like on the screen. Be creative. Take screen shots, pictures, images,etc.
c. Why can you not prove an identity using the graphing or tables tools? Explain.