If sin A=3/5 and cosB = 24/25 and angle A and angle B are both in quandrant 1. What is sthe exact value of cos (A-B)?

117/125

To find the exact value of cos(A-B), we can use the trigonometric identity:

cos(A-B) = cos(A)cos(B) + sin(A)sin(B)

Given that sin A = 3/5 and cos B = 24/25, we can substitute these values into the formula:

cos(A-B) = cos(A)cos(B) + sin(A)sin(B)
cos(A-B) = (cos(A))(24/25) + (sin(A))(3/5)

Now, to find the value of cos(A), we can use the Pythagorean identity:

cos^2(A) = 1 - sin^2(A)

Substituting the given value sin A = 3/5 into the Pythagorean identity:

cos^2(A) = 1 - (3/5)^2
cos^2(A) = 1 - (9/25)
cos^2(A) = 16/25

Taking the square root of both sides:

cos(A) = ±√(16/25)
cos(A) = ±(4/5)

Since angle A is in quadrant 1 (where cosine is positive), we take the positive value:

cos(A) = 4/5

Now substituting the values back into the formula for cos(A-B):

cos(A-B) = (4/5)(24/25) + (3/5)(3/5)
cos(A-B) = (96/125) + (9/25)

To add these fractions, we need a common denominator:

cos(A-B) = (96/125) * (1/1) + (9/25) * (5/5)
cos(A-B) = 96/125 + 45/125
cos(A-B) = 141/125

Therefore, the exact value of cos(A-B) is 141/125.

cos (A-B) = cosA cosB - sinA sinB

Note that cosA = 4/5 and sin B = 7/25, which you can prove using
cos^2 x + sin^2 x = 1