Hey, everyone. So far, we've been using our sum and difference identities in order to simplify and get exact values for trig functions of specific angles. But often, you'll be asked to use your sum and difference identities to work with expressions that have variables in them. So we're going to do that here. Doing this will allow you to simplify different expressions, so let's jump into this first one here. We have the sine of β + 30 degrees and we want to expand this expression using our sum and difference identities and then simplify.

Now here since I have the sine of θ + 30 degrees, I'm going to use my sum formula for sine. Expanding that out will give me the sine of θ times the cosine of 30 degrees. Then I'm adding that together with the cosine of θ times the sine of 30 degrees. The cosine and sine of 30 degrees are two values that I already know from the unit circle. The sine of θ and the cosine of θ are not things that I can simplify because they're just of a variable. But the cosine of 30 degrees is equal to the square root of 3⁄2, and the sine of 30 degrees is equal to 1⁄2.

So here I have the sine of θ times the square root of 3⁄2, plus the cosine of θ times 1⁄2. Now, this looks kind of weird, so let's go ahead and rearrange and simplify here. This is the square root of 3⁄2 times the sine of θ because we always want to have that coefficient in the front and then we're adding that together with 1⁄2 times the cosine of θ. Now we can simplify this a little further because these both have a denominator of 2. So I can rewrite this with my 1⁄2 on the outside factored out times the square root of 3 sine θ plus the cosine of θ. And we have simplified this as much as we can using our sum and difference formula.

Now let's look at another example. Here we have the cosine of π/4 - θ. Now here I want to use my difference formula for cosine. So using that to expand this out, this gives me the cosine of π/4 times the cosine of θ plus the sine of π/4 times the sine of θ. Again, we know these 2 trig values. The cosine and the sine of π/4 are actually both the square root of 2 over 2. So this gives me the square root of 2 over 2 times the cosine of θ plus the square root of 2 over 2 times the sine of θ.

Now, because these both have the same exact coefficient, I can go ahead and factor this out. Now, that will leave me with the square root of 2⁄2 times the cosine of θ plus the sine of θ. And that is my expanded out using the sum and difference formulas and then simplified as much as possible. Thanks for watching, and I'll see you in the next one.