Here's Your Angle Sum Identity for Sine (and Cosine)

I intend for this to be the last spit of trig proofs for a while. After this post, we'll have the tools we need to start working on derivatives. Specifically, we're going to work out how to expand and .

Once again, we're employing our now familiar unit circle:

You'll notice I have a fair amount of mark up on there. It seems messy, and there may vary well be an easier way to do it, but this was the way I did it, and it makes sense to me. So, we'll start like always with what we know about the problem:

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There are also some things we can infer about the diagram:

• is a right angle meaning

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Now we're ready to begin expanding our identities. Starting with :

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We'll start by finding the value for :

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• Simplifying, we get:
  

We're halfway there. Now we have to find :

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Simplifying again, we get:


So:


So, there's that. Time to move on to . Much like with , we'll be using a divide and conquer approach.

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So, starting with :

• Thinking back to our days of SOHCAHTOA we know that

• We also know that and (from our previous proof)
  

• So,

• Simplifying we get:

Again, we find ourselves halfway there. Now we need to find :

• We know

• We know from our previous example that

• This tells us

• Expanding using our previous proofs we get

• Simplifying we get
  

• Now we put it all together:
  

So, we can now use the following two identities:

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Thus, we close the book on trig for a while.

Pythagorean Theorem for Sines and Cosines

It's time to look at one of the most fundamental identities in trig:
I can't count the number of times I've used that identity. I can however, count the number of times I've used it since I knew why it was true. This proof will be a breeze after the SOHCAHTOA work. We'll use the unit circle again for this proof:

• To start with, we need to remember that with regards to right triangles when is the hypotenuse,

• Referring to the unit circle above, and specifically , we can say that and

• This means that

• So, using the Pythagorean Theorem, we can say that

• Giving us the end result