Find the exact value of sin(A+B) when angle A is arcsin(8/17) and angle B is arcsin(5/13)
To find the exact value of sin(A+B), where angle A is arcsin(8/17) and angle B is arcsin(5/13), we can use the following trigonometric identity:
sin(A + B) = sin A * cos B + cos A * sin B
First, let's find the values of sin A, sin B, cos A, and cos B:
Given:
sin A = 8/17
sin B = 5/13
We can use these values to find cos A and cos B using the Pythagorean identity:
cos^2 A = 1 - sin^2 A
cos^2 B = 1 - sin^2 B
cos A = sqrt(1 - sin^2 A)
cos B = sqrt(1 - sin^2 B)
Plugging in the values, we get:
cos A = sqrt(1 - (8/17)^2)
cos B = sqrt(1 - (5/13)^2)
Now, evaluate these expressions:
cos A = sqrt(1 - 64/289)
cos B = sqrt(1 - 25/169)
cos A = sqrt(289/289 - 64/289)
cos B = sqrt(169/169 - 25/169)
cos A = sqrt(225/289)
cos B = sqrt(144/169)
cos A = 15/17
cos B = 12/13
Now that we have the values of sin A, sin B, cos A, and cos B, we can substitute them into the formula for sin(A + B):
sin(A + B) = sin A * cos B + cos A * sin B
sin(A + B) = (8/17) * (12/13) + (15/17) * (5/13)
Evaluate this expression:
sin(A+B) = 96/221 + 75/221
Add the two fractions:
sin(A+B) = (96 + 75)/221
simplified:
sin(A+B) = 171/221
Therefore, the exact value of sin(A+B) when angle A is arcsin(8/17) and angle B is arcsin(5/13) is 171/221.