Answer: [tex]-\frac{240}{289}[/tex]
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Explanation:
Use the pythagorean trig identity [tex]\sin^2(\theta)+\cos^2(\theta) = 1[/tex] and plug in the fact that [tex]\cos(\theta) = \frac{8}{17}\\\\[/tex]
Isolating sine leads to [tex]\sin(\theta) = -\frac{15}{17}\\\\[/tex]. I'm skipping the steps here, but let me know if you need to see them.
The result is negative because we're in quadrant 4, when y < 0 so it's when sine is negative.
Therefore,
[tex]\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\\\\sin(2\theta) = 2*\left(-\frac{15}{17}\right)*\left(\frac{8}{17}\right)\\\\\sin(2\theta) = -\frac{240}{289}\\\\[/tex]
