if sin theta =-7/15 and theta is in the third quadrant then
Answers
Correct question :-
if sinθ = -7/15 and theta is in the third quadrant then find the value of
θ is in the third quadrant.
The value of
Using sin²θ + cos²θ = 1,
As θ is in third quadrant, so cosθ will be -24/25.
Cos θ is negative in third quadrant,
______________________
_______________________
_______________________
Therefore, the answer is 17/31.
Step-by-step explanation:
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} }
7cotθ+24tanθ
7cotθ−24tanθ
\underline{\underline{\sf{\color{magenta}{GIVEN:-}}}}
GIVEN:−
\bullet\ \sf{sin\theta=-\dfrac{7}{15} }∙ sinθ=−
15
7
\bullet∙ θ is in the third quadrant.
\underline{\underline{\sf{\color{magenta}{TO\ FIND:-}}}}
TO FIND:−
The value of
\sf{\longrightarrow \dfrac{7cot\theta-24tan\theta}{7cot\theta+24tan\theta} }⟶
7cotθ+24tanθ
7cotθ−24tanθ
\underline{\underline{\sf{\color{magenta}{SOLUTION:-}}}}
SOLUTION:−
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.
Cos θ is negative in third quadrant,
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\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
Therefore, the answer is 17/31.