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. You probably don't know the trig values of 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 to get to an answer much quicker.
Let's look at this first example. Here we have 75 degrees, and we want to rewrite that as the sum or difference of angles on our unit circle. I'm going to focus on angles in the first quadrant of the unit circle because those are the angles that I'm most familiar with. I'm going to consider 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. Looking at these angles, the first thing I notice is that if I have 30 degrees and add that together with 45 degrees, that gives me the 75 degrees I'm looking for. 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. Again, we want to find 2 out of these 3 angles that either add or subtract to that negative 15 degrees. Looking at these, I have my 45 degrees and my 30 degrees. If I subtract 45 from 30, that would give me negative 15. Here, if I took the cosine of 30 degrees minus 45 degrees, 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 \( \frac{7 \pi}{12} \). \( \frac{7 \pi}{12} \) is in radians, so this can be a little trickier to work with because it's a fraction. But remember, our angles are still these three angles: \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), and \( \frac{\pi}{3} \). We want to find 2 of these angles that add or subtract to give us \( \frac{7 \pi}{12} \). Looking at \( \frac{7 \pi}{12} \), I wanna think of numbers that add or subtract to 7. First thinking of 3 and 4, here, if I took \( \frac{3 \pi}{12} \) and added it together with \( \frac{4 \pi}{12} \), that gives me \( \frac{7 \pi}{12} \). But these don't look quite like these angles yet, but let's simplify these. \( \frac{3 \pi}{12} \) can be simplified to \( \frac{\pi}{4} \) and \( \frac{4 \pi}{12} \) can be simplified to \( \frac{\pi}{3} \), which are 2 of my 3 angles here. So here, I could take the cosine of \( \frac{\pi}{4} + \frac{\pi}{3} \), then use my sum formula to get an answer.
Earlier, we saw that we want to use our sum and difference identities whenever we see angles that are multiples of 15 degrees or \( \frac{\pi}{12} \) 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. \( \frac{7 \pi}{12} \) is a multiple of \( \frac{\pi}{12} \). 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 any questions.