If csc(ϴ) = 6, find the exact value if sin(2 ϴ)?

Question

1797 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-02T13:40:00

Please see below.

Without more information we can only answer

$sin 2 θ = \pm \frac{\sqrt{35}}{18}$ $sin2 theta = +-sqrt35/18$

Recall that $sin 2 θ = 2 sin θ cos θ$ $sin2 theta = 2sintheta cos theta$ .

So we want both $sin θ$ $sin theta$ and $cos θ$ $cos theta$ .

$csc θ = 6$ $csc theta = 6$ tells us that $\frac{1}{sin θ} = 6$ $1/sin theta = 6$

so that $sin θ = \frac{1}{6}$ $sin theta = 1/6$ .

Now, using ${sin}^{2} θ + {cos}^{2} θ = 1$ $sin^2 theta +cos^2 theta = 1$ , we get

$cos θ = \pm \frac{\sqrt{35}}{6}$ $cos theta = +-sqrt35/6$

Therefore,

$sin 2 θ = 2 sin θ cos θ$ $sin2 theta = 2sintheta cos theta$ .

$= 2 (\frac{1}{6}) (\pm \frac{\sqrt{35}}{6})$ $= 2(1/6)(+-sqrt35/6)$

$= \pm \frac{\sqrt{35}}{18}$ $= +- sqrt35/18$