SOLUTION: Given cosα = 1/2, -pi/2 < α < 0; sinβ=1/3, 0 < β < pi/2. Find cos(α + β)
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Question 615521: Given cosα = 1/2, -pi/2 < α < 0; sinβ=1/3, 0 < β < pi/2. Find cos(α + β)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Since it's easier to type, I'm going to use A and B instead of alpha and beta.
cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
We've been given cos(A) and sin(B). We need sin(A) and cos(B).
We can find sin(A) from cos(A) using:
Substituting in the given value for cos(A) we get:
which simplifies as follows:
Now we will find the square root of each side. Since we are told that A is between and 0 it must terminate in the 4th quadrant. Since sin is negative in the 4th quadrant, we know to use the negative square root:
which simplifies:
We can use a similar process to find cos(B):
Since B is between 0 and it must terminate in the 1st quadrant. cos is positive in the first quadrant so we will use the positive square root:
Now that we have all the values we need, we can go back to:
cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
and substitute in the values we have:
which simplifies as follows:
This may an acceptable answer. If not, then add the terms together:
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