Question

If tanθ=1/√3 find the value of sin θ

Answer 1

SOLUTION

GIVEN

$\displaystyle \sf{ \tan \theta = \frac{1}{ \sqrt{3} } }$

TO DETERMINE

$\displaystyle \sf{ \sin \theta }$

EVALUATION

USING GENERAL SOLUTION

$\displaystyle \sf{ \tan \theta = \frac{1}{ \sqrt{3} } }$

$\implies \displaystyle \sf{ \tan \theta = \tan \frac{\pi}{6} }$

$\displaystyle \sf{ \theta = n\pi + \frac{\pi}{6} } \: \: where \: n \: = 0, \pm \: 1,\pm 2,....$

CASE : 1

$\sf{ \underline{If \: \: \theta \: \: is \: in \: 1st \: quadrant} : }$

Then

$\displaystyle \sf{ \sin\theta = sin \frac{\pi}{6} } \:$

$\implies \displaystyle \sf{ \sin\theta = \frac{1}{2} } \:$

CASE : 2

$\sf{ \underline{If \: \: \theta \: \: is \: in \: 3rd \: quadrant} : }$

$\displaystyle \sf{ \sin\theta = - sin \frac{\pi}{6} } \:$

$\implies\displaystyle \sf{ \sin\theta = - \frac{1}{2} } \:$

USING PRINCIPAL VALUE

$\displaystyle \sf{ \tan \theta = \frac{1}{ \sqrt{3} } }$

$\implies \displaystyle \sf{ \tan \theta = \tan \frac{\pi}{6} }$

$\implies \displaystyle \sf{ \theta = \frac{\pi}{6} }$

So

$\displaystyle \sf{ \sin \theta }$

$\displaystyle \sf{ = \sin \frac{\pi}{6} }$

$= \displaystyle \sf{ \frac{1}{2} }$

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