Question

if sin theta =-7/15 and theta is in the third quadrant then $\sf\dfrac{7cot\theta-24tan\theta}{7cot\theta +24tan\theta} =$​

Answer 1

Answer:

$\frac{7cot\beta-24tan\beta}{7cot\beta+24tan\beta}$ = $\frac{1}{43}$

Step-by-step explanation:

Given: sinβ = -7/15 and β is in third quadrant.

⇒ cotβ = 4√11/7

⇒ 7cotβ = 7×4√11/7 = 4√11

⇒ tanβ = 7/4√11

⇒ 24tanβ = 24*7/4√11 = 42/√11

⇒ $\frac{7cot\beta-24tan\beta}{7cot\beta+24tan\beta}$ = $\frac{4\sqrt{11}-\frac{42}{\sqrt{11}}}{4\sqrt{11}+\frac{42}{\sqrt{11}}}$ = $\frac{4*11-42}{4*11+42}$ = $\frac{44-42}{44+42}$ = $\frac{2}{86}$

⇒ $\frac{7cot\beta-24tan\beta}{7cot\beta+24tan\beta}$ = $\frac{1}{43}$

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Answer 2

Correct question :-

if sinθ  = -7/15 and theta is in the third quadrant then find the value of

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

$\underline{\underline{\sf{\color{magenta}{GIVEN:-}}}}$

$\bullet\ \sf{sin\theta=-\dfrac{7}{15} }$

$\bullet$ θ is in the third quadrant.

$\underline{\underline{\sf{\color{magenta}{TO\ FIND:-}}}}$

The value of

$\sf{\longrightarrow \dfrac{7cot\theta-24tan\theta}{7cot\theta+24tan\theta} }$

$\underline{\underline{\sf{\color{magenta}{SOLUTION:-}}}}$

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.

• Cos θ is negative in third quadrant,

______________________

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

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

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

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

_______________________

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

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

_______________________

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

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

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

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

Therefore, the answer is 17/31.