Question

if sun $\sf \theta = -\dfrac{7}{25}\: and \theta \:is$ in the third quadrant ,then $\dfrac{7cot \theta -24tan \theta }{7cot \theta +24 tan \theta }$=​

Answer 1

Answer :-

$\displaystyle \bf{ \frac{17}{31} }$

Step-by-step explanation :-

$\sf{sin\theta=-\dfrac{7}{25} }$

Using sin²θ + cos²θ = 1,

$\displaystyle \sf{\bigg(\frac{-7}{25}\bigg)^2 +cos^2\theta=1}$

$\implies\displaystyle \sf{cos^2\theta=1-\frac{49}{625} }$

$\implies\displaystyle \sf{cos^2\theta=\frac{625}{625}-\frac{49}{625} }$

$\displaystyle \sf{\implies cos^2\theta=\frac{576}{625} }$

$\displaystyle \sf{\implies cos\theta=\pm\frac{24}{25} }$

As θ is in third quadrant, so cosθ will be -24/25.

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$\displaystyle \bf{ tan\theta=\frac{sin\theta}{cos\theta} }$

$\displaystyle \sf{\implies tan\theta=\frac{-7}{25}\div \frac{-24}{25} }$

$\displaystyle \sf{\implies tan\theta=\frac{-7}{25}\times\frac{25}{-24} }$

$\displaystyle \sf{\implies tan\theta=\frac{7}{24} }$

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$\displaystyle \bf{ cot\theta=\frac{1}{tan\theta}}$

$\displaystyle \sf{\implies cot\theta=\frac{24}{7} }$

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$\displaystyle \sf{\frac{7cot\theta-24tan\theta}{7cot\theta+24tan\theta} }$

$\displaystyle \sf{= \frac{7\times\frac{24}{7}-24\times\frac{7}{24} }{7\times\frac{24}{7}+24\times\frac{7}{24} } }$

$\displaystyle \sf{= \frac{24-7}{24+7} }$

$\displaystyle \sf{= \frac{17}{31} }$