if a and b are acute angles, sin(a+b)= 56/65, and sin b = 5/13, find sin a
To find sin(a), we can use the identity: sin(a+b) = sin(a)cos(b) + cos(a)sin(b).
Given that sin(a+b) = 56/65 and sin(b) = 5/13, we can substitute these values into the formula:
56/65 = sin(a)cos(b) + cos(a)sin(b)
Now, we need to find the values of cos(a) and cos(b).
Since a and b are both acute angles, we know that sin(a) and sin(b) are both positive. Therefore, cos(a) and cos(b) are both nonnegative.
We also know that sin^2(a) + cos^2(a) = 1 and sin^2(b) + cos^2(b) = 1. Using these identities, we can solve for cos(a) and cos(b):
sin^2(a) = 1 - cos^2(a)
cos^2(a) = 1 - sin^2(a)
sin^2(b) = 1 - cos^2(b)
cos^2(b) = 1 - sin^2(b)
Given that sin(b) = 5/13, we can substitute this value into the formula for cos^2(b):
cos^2(b) = 1 - (5/13)^2
cos^2(b) = 1 - 25/169
cos^2(b) = 144/169
cos(b) = sqrt(144/169)
cos(b) = 12/13
Now, we can substitute the values of sin(b) and cos(b) into the equation for sin(a+b):
56/65 = sin(a)(12/13) + cos(a)(5/13)
Multiplying both sides of the equation by 13:
728/65 = sin(a)(12) + cos(a)(5)
Rearranging the equation:
sin(a)(12) + cos(a)(5) = 728/65
Since sin(a) and cos(a) are both nonnegative, we can divide both sides of the equation by 13:
56/5 = sin(a)(12/13) + cos(a)(5/13)
Since sin(a) and cos(a) are both nonnegative, we can multiply both sides of the equation by 13/12:
(56/5)(13/12) = sin(a) + cos(a)(5/12)
Simplifying the equation:
(56/5)(13/12) = sin(a) + (5/12)cos(a)
Now, we can solve for sin(a):
1696/60 = sin(a) + (5/12)cos(a)
Simplifying the equation:
848/60 = sin(a) + (5/12)cos(a)
Dividing both sides of the equation by 4/5:
(848/60)/(4/5) = sin(a) + (5/12)cos(a)
Simplifying the equation:
1060/12 = sin(a) + (5/12)cos(a)
Subtracting (5/12)cos(a) from both sides of the equation:
1060/12 - (5/12)cos(a) = sin(a)
Simplifying the equation:
(1060 - 5cos(a))/12 = sin(a)
Therefore, sin(a) = (1060 - 5cos(a))/12.