12. Trigonometric Identities
Sum and Difference Identities
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Hey everyone. In this problem, we're asked to rewrite each argument as the sum or difference of 2 angles on the unit circle. Now this would be helpful if you were asked to find the exact value of an angle, like say the sine of 75 degrees. Now you probably don't know the trig values of the 75 degrees off the top of your head. So if instead we rewrite this in terms of angles that we do know on the unit circle, we can then use a sum or difference formula in order to get to an answer much quicker. So let's look at this first example. Here we have 75 degrees, and we wanna rewrite that as the sum or difference of angles on our unit circle. Now I'm going to focus on angles that are in the first quadrant of the unit circle because those are the angles that I'm the most familiar with. So here I'm going to look at 30 degrees, 45 degrees, and 60 degrees, and I want to find some combination of 2 of these angles that add or subtract to 75 degrees. Now looking at these angles, the first thing that I notice is that if I have 30 degrees and I add that together with 45 degrees, that gives me the 75 degrees that I'm looking for. So here, if I took the sine of 30 degrees plus 45 degrees and then used a sum formula, I could get to my answer. Now let's look at a second example. Here we have the cosine of negative 15 degrees. So, again, we want to find 2 out of these 3 angles that either add or subtract to that negative 15 degrees. Now looking at these, I have my 45 degrees and my 30 degrees. And if I subtracted 45 from 30, that would give me negative 15. So here, if I took 30 degrees, a minus 45 degrees, and I took the cosine of that, and then used a difference formula, I could then get to an answer. Now let's look at one final example. Here we have the cosine of 7 pi over 12. Now 7 pi over 12 is in radians, so this can be a little bit trickier to work with because it's fraction. But remember that our angles are still these three angles. We have pi over 6, pi over 4, and pi over 3. So we wanna find 2 of these angles that add or subtract to give us 7 pi over 12. Now looking at 7 pi over 12, I wanna be thinking of numbers that either add or subtract to that. Right? So if I think of that 7 and I think of 2 numbers that add or subtract to 7, I first think of 3 and 4. Now here, if I took 3 pi over 12 and I added it together with 4 pi over 12, that gives me 7 pi over 12. But these don't look quite like these angles yet, but let's simplify these. 3 pi over 12 can be simplified to pi over 4 and 4 pi over 12 can be simplified to pi over 3, which are 2 of my 3 angles here. So here I could take the cosine of pi over 4 plus pi over 3, then use my sum formula to get an answer. Now earlier we saw that we want to use our sum and difference identities whenever we see angles that are multiples of 15 degrees or pi over 12 radians, and that's exactly what we see here. Here we have the sine of 75 degrees. 75 is a multiple of 15, as is negative 15. 7 pi over 12 is a multiple of pi over 12. So whenever we see that, we want to go ahead and break down our argument as a sum or difference and then use our identities from there. Thanks for watching, and let me know if you have questions.