Evaluate the expression under the given conditions. (show work please)
sin(A-B); tan A=12/5, A in quadrant 3, sinB=-3sqrt(10)/10, B in quadrant 4.
To evaluate the expression sin(A-B) given the conditions tan A = 12/5 with A in quadrant 3 and sin B = -3sqrt(10)/10 with B in quadrant 4, we can use the trigonometric identity:
sin(A-B) = sin A * cos B - cos A * sin B
First, let's find the values of cos A and cos B.
Since tan A = 12/5 and A is in quadrant 3, we can use the Pythagorean identity tan^2 A + 1 = sec^2 A to find the value of sec A (which is equal to 1/cos A):
tan^2 A + 1 = sec^2 A
(12/5)^2 + 1 = sec^2 A
144/25 + 1 = sec^2 A
169/25 = sec^2 A
√(169/25) = √(sec^2 A)
13/5 = sec A
Since A is in quadrant 3, cos A is negative, and therefore:
cos A = -5/13
Now, let's find the value of cos B using the Pythagorean identity sin^2 B + cos^2 B = 1:
sin^2 B + cos^2 B = 1
(-3sqrt(10)/10)^2 + cos^2 B = 1
9/10 + cos^2 B = 1
cos^2 B = 1 - 9/10
cos^2 B = 1/10
cos B = ±√(1/10)
Since B is in quadrant 4, cos B is positive, and therefore:
cos B = √(1/10)
Now we have the values of sin A, sin B, cos A, and cos B, so we can evaluate the expression sin(A-B):
sin(A-B) = sin A * cos B - cos A * sin B
= (12/5) * (√(1/10)) - (-5/13) * (-3sqrt(10)/10)
To simplify further, we can rationalize the denominator of √(1/10):
√(1/10) = √(1)/√(10) = 1/√(10) = √(10)/10
Plugging in the simplified values:
sin(A-B) = (12/5) * (√(10)/10) - (-5/13) * (-3sqrt(10)/10)
= (12√(10))/50 + (15√(10))/130
Combining the terms with the common denominator:
sin(A-B) = (12√(10))/50 + (15√(10))/130
= (12√(10))(13)/(50)(13) + (15√(10))(5)/(130)(5)
= 156√(10)/650 + 75√(10)/650
= (156√(10) + 75√(10))/650
= (231√(10))/650
Therefore, sin(A-B) = (231√(10))/650.