Suppose cosA = 12/13 with 0º≤A≤90º. Suppose also that sinB = 8/17 with 90º≤B≤180º. Find cos(A - B).
See previous post: Tue, 10-14-14, 2:39 PM.
To find cos(A - B), we can use the formula:
cos(A - B) = cosA * cosB + sinA * sinB
Given that cosA = 12/13 and sinB = 8/17, we need to find the values of cosB and sinA in order to calculate cos(A - B).
To find cosB, we can use the identity:
sin^2B + cos^2B = 1
Given that sinB = 8/17, we can solve for cosB:
cos^2B = 1 - sin^2B
cos^2B = 1 - (8/17)^2
cos^2B = 1 - 64/289
cos^2B = (289 - 64) / 289
cos^2B = 225 / 289
cosB = ±√(225 / 289)
Since B is in the range 90º ≤ B ≤ 180º, the value of cosB is negative. Therefore, we have:
cosB = -√(225 / 289) = -15/17
Now, to find sinA, we can use the identity:
sin^2A + cos^2A = 1
Given that cosA = 12/13, we can solve for sinA:
sin^2A = 1 - cos^2A
sin^2A = 1 - (12/13)^2
sin^2A = 1 - 144/169
sin^2A = (169 - 144) / 169
sin^2A = 25 / 169
sinA = ±√(25 / 169)
Since A is in the range 0º ≤ A ≤ 90º, the value of sinA is positive. Therefore, we have:
sinA = √(25 / 169) = 5/13
Now that we have the values of cosB = -15/17 and sinA = 5/13, we can calculate cos(A - B):
cos(A - B) = cosA * cosB + sinA * sinB
cos(A - B) = (12/13) * (-15/17) + (5/13) * (8/17)
cos(A - B) = (-180/221) + (40/221)
cos(A - B) = -140/221
Therefore, cos(A - B) is equal to -140/221.