Final answer:
To find the exact solution of sin(2θ), we can use the double-angle formula for sine. Given that sin(θ) = -7/25 and π < θ < 3π/2, the exact solution of sin(2θ) is -336/625.
Step-by-step explanation:
To find the exact solution of sin(2θ), we can use the double-angle formula for sine.
The double-angle formula for sine is: sin(2θ) = 2sin(θ)cos(θ)
Given that sin(θ) = -7/25 and π < θ < 3π/2, we can substitute these values into the formula.
sin(2θ) = 2(-7/25)cos(θ)
Now, we need to find the value of cos(θ). Since sin(θ) = -7/25 and we know that π < θ < 3π/2, we are in the third quadrant where cosine is negative. Using the Pythagorean identity sin^2(θ) + cos^2(θ) = 1, we can find cos(θ) as follows:
cos(θ) = sqrt(1 - sin^2(θ)) = sqrt(1 - (-7/25)^2) = sqrt(1 - 49/625) = sqrt(576/625) = 24/25
Substituting back into the formula for sin(2θ), we have:
sin(2θ) = 2(-7/25)(24/25) = -336/625
Therefore, the exact solution of sin(2θ) is -336/625.