Answer:
16/65
Explanation:
You want sin(2a) if ((2cos(a)+3sin(a))/(3cos(a)-2sin(a)) = -2.
Solution
Let c = cos(a) and s = sin(a) to help reduce the writing.
We are given ...
(2c +3s)/(3c -2s) = -2
Multiplying by the denominator, we have ...
2c +3s = -2(3c -2s)
2c +3s = -6c +4s . . . . eliminate parentheses
8c = s . . . . . . . . . . . . . add 6c-3s
s/c = 8 = tan(a) . . . . . . divide by c
At this point, you can use a calculator to find sin(2a) from ...
sin(2a) = sin(2·arctan(8)) = 16/65
You can also get there using the identity for sin(2a):
sin(2a) = 2sin(a)cos(a) = 2sin(a)cos²(a)/cos(a) = 2tan(a)·cos²(a)
= 2tan(a)/sec²(a) = 2tan(a)/(1+tan²(a))
Using the found value of tan(a), this is ...
sin(2a) = 2(8)/(1+8²) = 16/65