Question 890077: Find the exact value:
cos75-cos15
Answer by alakazam2192(3)   (Show Source):
You can put this solution on YOUR website! Two important trigonometric identities say this:
1. cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
2. cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
We will use these to solve for what cos75 and cos15 are. Let's start with cos75 first.
To properly use the above identities, we should think of two numbers a and b that are easy identities for us to use. For cos75, the two numbers that will be the most useful are 45 and 30. Not only do these add up to 75, but also, cos45, cos30, sin45, and sin30 are easy to evaluate, due to them being easy identities. If these said identities aren't too familiar, I would suggest reviewing them, as knowing these is crucial to problems like this.
Continuing on, identity 1, which is represented above, yields:
cos(45)cos(30) - sin(45)sin(30) = cos(75)
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Now, let's do cos15. Here, we can still use 45 and 30 as our numbers, since subtracting them gets us 15. However, we must use identity 2 to help us this time:
cos(45)cos(30) + sin(45)sin(30) = cos(15)
Therefore, we have:
cos(75) - cos(15) = [cos(45)cos(30) - sin(45)sin(30)]-[cos(45)cos(30) + sin(45)sin(30)] = -2sin(45)sin(30)
Using other trigonometric identities, we can say that sin(45) = sqrt(2)/2, and sin(30) = 1/2. Therefore, we obtain:
-2(sqrt(2)/2)*(1/2) = -sqrt(2)/2
I hope this helped!
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