if a and b are acute angles, sin(a+b)= 56/65, and sin b = 5/13, find sin a
sin a= 46/65
To find sin(a), we can use the sum of angles identity for sine:
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 plug these values into the equation:
56/65 = sin(a)cos(b) + cos(a)(5/13)
Now, let's solve for cos(a) using the Pythagorean identity:
cos^2(a) = 1 - sin^2(a)
Since a is an acute angle, sin(a) and cos(a) are both positive. Thus, we have:
cos(a) = sqrt(1 - sin^2(a))
Now we can substitute this expression for cos(a) back into our original equation:
56/65 = sin(a)sqrt(1 - (5/13)^2) + (5/13)sqrt(1 - sin^2(a))
Simplifying this equation will allow us to solve for sin(a).