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 trigonometric identity:
cos(A + B) = cosA * cosB - sinA * sinB
Given that sinA = 12/13 and sinB = -7/25, we need to find cosA and cosB.
We know that sin^2(A) + cos^2(A) = 1, so we can find cosA using this equation:
sin^2(A) + cos^2(A) = 1
(12/13)^2 + cos^2(A) = 1
144/169 + cos^2(A) = 1
cos^2(A) = 25/169
cosA = ±√(25/169)
cosA = ±(5/13)
Since A is in the second quadrant (90º to 180º), the cosine of A will be negative.
Therefore, cosA = -5/13.
Similarly, we can find cosB:
sin^2(B) + cos^2(B) = 1
(-7/25)^2 + cos^2(B) = 1
49/625 + cos^2(B) = 1
cos^2(B) = 576/625
cosB = ±√(576/625)
cosB = ±(24/25)
Since B is in the fourth quadrant (-90º to 0º), the cosine of B will also be positive.
Therefore, cosB = 24/25.
Now we can substitute the values of cosA = -5/13 and cosB = 24/25 into the formula for cos(A + B):
cos(A + B) = cosA * cosB - sinA * sinB
= (-5/13) * (24/25) - (12/13) * (-7/25)
= -120/325 + 84/325
= -36/325
Therefore, cos(A + B) = -36/325.
To find cos(A + B), we need to use the trigonometric identity for the cosine of the sum of two angles.
The identity is: cos(A + B) = cosA * cosB - sinA * sinB
Given that sinA = 12/13 and sinB = -7/25, we need to find the values of cosA and cosB.
To find cosA, we can use the Pythagorean identity: cos^2(A) + sin^2(A) = 1
Using this identity, we can calculate cosA:
cos^2(A) = 1 - sin^2(A)
cos^2(A) = 1 - (12/13)^2
cos^2(A) = 1 - 144/169
cos^2(A) = (169 - 144)/169
cos^2(A) = 25/169
Taking the square root of both sides, we get:
cos(A) = ±√(25/169)
cos(A) = ±(5/13)
Since 90º ≤ A ≤ 180º, A is in the second quadrant where cosine is negative. Therefore, cos(A) = -5/13.
Similarly, we can find cosB:
cos^2(B) = 1 - sin^2(B)
cos^2(B) = 1 - (-7/25)^2
cos^2(B) = 1 - 49/625
cos^2(B) = (625 - 49)/625
cos^2(B) = 576/625
Taking the square root of both sides, we get:
cos(B) = ±√(576/625)
cos(B) = ±(24/25)
Since -90º ≤ B ≤ 0º, B is in the fourth quadrant where cosine is positive. Therefore, cos(B) = 24/25.
Now that we have cosA = -5/13 and cosB = 24/25, we can find cos(A + B):
cos(A + B) = cosA * cosB - sinA * sinB
cos(A + B) = (-5/13) * (24/25) - (12/13) * (-7/25)
cos(A + B) = (-5 * 24)/(13 * 25) - (12 * 7)/(13 * 25)
cos(A + B) = -120/325 + 84/325
cos(A + B) = -36/325
So, cos(A + B) = -36/325.