Final answer:
To find sin2(x), we determined that x is in the second quadrant, used the Pythagorean identity to find sin(x) = (15/17), and then applied the double-angle formula to obtain sin(2x) = -240/289.
Step-by-step explanation:
To find sin2(x) when given cos(x) = -(8/17) and tan(x) < 0, we first determine the quadrant in which the angle x lies. Since the cosine is negative and tangent is negative, x must lie in the second quadrant where both sine and cosine have opposite signs (sine is positive and cosine is negative). In this quadrant, sin(x) will be positive.
Using the Pythagorean identity, sin^2(x) + cos^2(x) = 1, we can solve for sin(x).
sin^2(x) = 1 - cos^2(x)
sin^2(x) = 1 - (-8/17)^2
sin^2(x) = 1 - (64/289)
sin^2(x) = (289/289) - (64/289)
sin^2(x) = 225/289
Therefore, sin(x) = ±(15/17), but since we are in the second quadrant, sin(x) is positive:
sin(x) = (15/17)
Now we use the double-angle formula for sine: sin(2x) = 2sin(x)cos(x).
sin(2x) = 2 * (15/17) * (-8/17)
sin(2x) = 2 * (-120/289)
sin(2x) = -240/289