Suppose sinA = 12/13 with 90º≤A≤180º. Suppose also that sinB = -7/25 with -90º≤B≤0º. Find cos(A - B).
To find cos(A - B), we can use the following trigonometric identity:
cos(A - B) = cosA * cosB + sinA * sinB
Given that sinA = 12/13 and sinB = -7/25, we can find cosA and cosB using the Pythagorean identity:
cosA = √(1 - sin²A)
cosB = √(1 - sin²B)
Firstly, let's find cosA:
sinA = 12/13
cosA = √(1 - (sinA)²)
= √(1 - (12/13)²)
= √(1 - 144/169)
= √(169/169 - 144/169)
= √(25/169)
= 5/13
Next, let's find cosB:
sinB = -7/25
cosB = √(1 - (sinB)²)
= √(1 - (-7/25)²)
= √(1 - 49/625)
= √(625/625 - 49/625)
= √(576/625)
= 24/25
Now we can substitute the values of cosA and cosB into the formula for cos(A - B):
cos(A - B) = cosA * cosB + sinA * sinB
= (5/13) * (24/25) + (12/13) * (-7/25)
= (5 * 24 + 12 * -7) / (13 * 25)
= (120 - 84) / 325
= 36/325
So, cos(A - B) = 36/325.