How do you use $sin θ = \frac{1}{3}$ $sintheta=1/3$ to find $tan θ$ $tantheta$ ?

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How do you use $sin θ = \frac{1}{3}$ $sintheta=1/3$ to find $tan θ$ $tantheta$ ?

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Answer 1 · 2017-01-08T15:45:28

$\pm \frac{\sqrt{2}}{4}$ $+- sqrt2/4$

Explanation:

First find cos t.
${cos}^{2} t = 1 - {sin}^{2} t$ $cos^2 t = 1 - sin^2 t$
${cos}^{2} t = 1 - \frac{1}{9} = \frac{8}{9}$ $cos ^2 t = 1 - 1/9 = 8/9$
$cos t = \pm \frac{2 \sqrt{2}}{3}$ $cos t = +- (2sqrt2)/3$
$tan t = \frac{sin t}{cos t} = \pm (\frac{1}{3}) (\frac{3}{2 \sqrt{2}}) = \pm \frac{1}{2 \sqrt{2}} = \pm \frac{\sqrt{2}}{4}$ $tan t = sin t/(cos t) = +- (1/3)(3/(2sqrt2)) = +- 1/(2sqrt2) = +- sqrt2/4$
Note about the signs of tan t.
On the unit circle:
sin t = 1/3 --> cos t either + or - , there for --> tan t either + or -
Check by calculator.
$sin t = \frac{1}{3}$ $sin t = 1/3$ --> $t = 19^{\circ} 47$ $t = 19^@47$ and $t = 180 - 19.47 = 160^{\circ} 53$ $t = 180 - 19.47 = 160^@53$
tan 19.47 = 0.35 and tan 160.53 = - 0.35
$\pm \frac{\sqrt{2}}{4} = \pm \frac{1.414}{5} = \pm 0.35$ $+- sqrt2/4 = +- 1.414/5 = +- 0.35$ OK

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