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

$\huge\boxed{\fcolorbox{purple}{ink}{Answer}}$

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

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.

_____________________

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

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

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

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

______________________

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

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

______________________

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

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

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

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

Answer 2

\huge\boxed{\fcolorbox{purple}{ink}{Answer}}

Answer

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

31

17

Step-by-step explanation :-

\sf{sin\theta=-\dfrac{7}{25} }sinθ=−

25

7

Using sin²θ + cos²θ = 1,

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

25

−7

)

2

+cos

2

θ=1

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

2

θ=1−

625

49

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

2

θ=

625

625

−

625

49

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

2

θ=

625

576

\displaystyle \sf{\implies cos\theta=\pm\frac{24}{25} }⟹cosθ=±

25

24

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

_____________________

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

cosθ

sinθ

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

25

−7

÷

25

−24

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

25

−7

×

−24

25

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

24

7

______________________

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

tanθ

1

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

7

24

______________________

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

7cotθ+24tanθ

7cotθ−24tanθ

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

7×

7

24

+24×

24

7

7×

7

24

−24×

24

7

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

24+7

24−7

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

31

17