To find the exact value of sin(2a), we can use the double angle formula for sine:
sin(2a) = 2sin(a)cos(a)
We are given that cot(a) = 3, which is the reciprocal of tangent (tan(a)). Since tangent is negative in the third quadrant, we can determine that tan(a) = -1/3.
Using the Pythagorean identity for tangent:
tan^2(a) + 1 = sec^2(a)
(-1/3)^2 + 1 = sec^2(a)
1/9 + 1 = sec^2(a)
10/9 = sec^2(a)
Taking the square root of both sides:
√(10/9) = sec(a)
Since secant is the reciprocal of cosine, we have:
sec(a) = 3/√10
Now, we can substitute the values into the double-angle formula for sine:
sin(2a) = 2sin(a)cos(a)
= 2(sin(a) * 1/√10) * (3/√10)
= 6sin(a)/10
Since sin(a) is not given directly, we need to find it. We are given that sin(b) = 15/17, where b lies in the second quadrant. Using the Pythagorean theorem:
sin^2(b) + cos^2(b) = 1
(15/17)^2 + cos^2(b) = 1
225/289 + cos^2(b) = 1
cos^2(b) = 289/289 - 225/289
cos^2(b) = 64/289
Taking the square root of both sides:
cos(b) = ±√(64/289)
cos(b) = ±8/17
Since b lies in the second quadrant, cosine is positive, so cos(b) = 8/17.
Using the Pythagorean identity for sine:
sin^2(b) + cos^2(b) = 1
(15/17)^2 + (8/17)^2 = 1
225/289 + 64/289 = 1
289/289 = 1
This verifies that sin(b) = 15/17 and cos(b) = 8/17.
Now, we can find sin(a) using the following relation between the sine and cosine of complementary angles:
sin(a) = cos(b)
Therefore, sin(a) = 8/17.
Substituting this value into the expression for sin(2a):
sin(2a) = 6sin(a)/10
= 6(8/17)/10
= 48/170
= 24/85
Therefore, the exact value of sin(2a) is 24/85.