Answer: The value of sin(2x) can be found using the identity:
sin(2x) = 2sin(x)cos(x)
Since tan(x) = -1/5, we can use the tangent definition to find the sine and cosine values:
tan(x) = sin(x) / cos(x) = -1/5
Cross multiplying and solving for sin(x), we get:
sin(x) = -5 / sqrt(26)
And using the Pythagorean identity:
cos^2(x) + sin^2(x) = 1
We can find cos(x):
cos(x) = sqrt(1 - sin^2(x)) = sqrt(1 - (-5/sqrt(26))^2) = sqrt(26) / sqrt(26) = sqrt(26) / sqrt(26) = sqrt(26) / 5
Finally, substituting the values of sin(x) and cos(x) into the formula for sin(2x), we get:
sin(2x) = 2sin(x)cos(x) = 2 * (-5 / sqrt(26)) * (sqrt(26) / 5) = -2 sqrt(26) / 5 = -2 sqrt(26) / 5 = -2 sqrt(26) / 5
So the exact value of sin(2x) is -2 sqrt(26) / 5.
Explanation: