Question

Given that cos(theta) = 8/17 and that theta lies in Quadrant IV, what is the exact value of sin 2(theta)?

Answer 1

Answer:
`-(240)/(289)`

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Step-by-step explanation:

Use the pythagorean trig identity
`\sin^2(\theta)+\cos^2(\theta) = 1` and plug in the fact that
`\cos(\theta) = (8)/(17)\\\\`

Isolating sine leads to
`\sin(\theta) = -(15)/(17)\\\\`. 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,

`\sin(2\theta) = 2\sin(\theta)\cos(\theta)\\\\\sin(2\theta) = 2*\left(-(15)/(17)\right)*\left((8)/(17)\right)\\\\\sin(2\theta) = -(240)/(289)\\\\`