Trig+Indentities+9-29

Summary: The two new identities we learned in class today were the Angle Sum and Angle Difference Identities. We proved how each identity works and we discovered how we can use these new identities to find exact measures of more angles (other than the three in each quadrant that we already learned this year).

__Angle Sum Identities__
The Angle Sum Identities involves sine, cosine, and tangent. We learned about the sine and the cosine identities first. Mrs. Tyson wrote out the identities for us, which are the following:

sin(α + β) = sin α cos β + cos α sin β cos(α + β) = cos α cos β − sin α sin β

Just as with every other identity we've talked about, we worked through the proof of the identity so that we all had a clear understanding of why this identity works. It's one thing to have it memorized, but it's another thing to understand how it works and be able to figure out missing pieces if you, at some point, draw a blank on what the identity is.

Speaking of memorizing things, Mrs. Tyson gave some examples of how to remember these identities. You can form your own unique way to memorize these, it doesn't have to be any certain way. Mrs. Tyson just highly stressed that you should store these identities somewhere in your mind, because they will come in handy (like on the next test!)

The equation for the tangent angle sum identity is as follows:



The proof for this equation is very long and lengthy. The basic outline of the process is to first turn all the tangents into something more manageable, like sine and cosine! From there, the next step is to find common denominators, cancel like terms, and use your knowledge of algebra to solve the rest! You do not need to know the proof, however, you should memorize this equation, as well as the ones involving sine and cosine.

__Angle Difference Identities__
The Angle Difference Identities involves sine, cosine, and tangent as well. The major difference between the Angle //Sum// Identities and the Angle //Difference// Identities, is not only their names, but the sign inside the parenthesis. If you look at the Angle Sum Identities, you'll notice that between alpha and beta inside the parenthesis is a plus every single time. That is why they are called Angle __Sum__ Identities (Sum = +)

The Angle Difference formulas all have minus signs inside the parenthesis. That is why they are called the Angle __Difference__ Identities (Difference = -).

The following are the Angle Difference Identities:

sin(α − β) = sin α cos β − cos α sin β cos(α − β) = cos α cos β + sin α sin β

All of these have similar forms in comparison to the Angle Sum Identities, but they differ in the **signs** within the equations.

Once again, Mrs. Tyson stressed the importance of memorizing these identities! Learn them and remember them!

__What do I do with these equations?__
These identities help in a very specific way. Currently, before learning these identities, we only knew three angles in each quadrant (the angles with denominator 3, 4, and 6). With these new identities, we are able to find the exact measures of other angles in any quadrant.