If Sinθ = 1/3, what is CosθSin2θ?

Question

Using Sin^2 + cos^2 = 1

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Answer 1 · 2018-05-03T00:24:41

$\frac{16}{27}$ $16/27$

To find $cos θ$ $costheta$

$cos θ = \pm \sqrt{1 - {sin}^{2} θ}$ $costheta= +-sqrt(1-sin^2theta)$

$cos θ = \pm \sqrt{1 - {(\frac{1}{3})}^{2}}$ $costheta= +-sqrt(1-(1/3)^2)$

$cos θ = \pm \frac{2 \sqrt{2}}{3}$ $costheta= +-(2sqrt2)/3$

Depends on what quadrant this angle is in determines whether the cosine value is negative or positive.

Apply sine double angle identity:
$cos θ \cdot 2 sin θ cos θ$ $costheta*2sinthetacostheta$

$(\pm \frac{2 \sqrt{2}}{3}) \cdot 2 (\frac{1}{3}) (\pm \frac{2 \sqrt{2}}{3}) =$ $(+-(2sqrt2)/3)*2(1/3)(+-(2sqrt2)/3)=$

$\frac{16}{27}$ $16/27$