Question

pls give a detailed solution for this ​

Answer 1

QUESTION:

$\sf Cot(60^{\circ}+m) = \dfrac{1}{3\sqrt{3}} \ find \ Cot(60^{\circ} - m)$

ANSWER:

$\sf Cot(60^{\circ} - m) = \dfrac{ - 2}{ \sqrt{3} }$

GIVEN:

$\sf Cot(60^{\circ}+m) = \dfrac{1}{3\sqrt{3}}$

TO FIND:

$\sf Cot(60^{\circ} - m)$

EXPLANATION:

$\boxed{ \bold{\large{ \gray{Cot(x + y) = \dfrac{Cot \ x \ Cot \ y - 1}{Cot \ x + Cot \ y}}}}}$

$\sf Cot(60^{\circ}+m) = \dfrac{Cot \ 60^{\circ} Cot \ m - 1}{Cot \ 60^{\circ} + Cot \ m}$

Cot 60° = 1/√3

$\sf Cot(60^{\circ}+m) = \dfrac{ \dfrac{Cot \ m}{ \sqrt{3} } - 1}{ \dfrac{1}{ \sqrt{3} } + Cot \ m}$

$\sf Cot(60^{\circ}+m) = \dfrac{\dfrac{Cot \ m - \sqrt{3}}{ \sqrt{3}}}{ \dfrac{1 + \sqrt{3} Cot \ m}{ \sqrt{3}}}$

$\sf Cot(60^{\circ}+m) = \dfrac{Cot \ m - \sqrt{3}}{ 1 + \sqrt{3} Cot \ m}$

$\sf Cot(60^{\circ}+m) = \dfrac{1}{3\sqrt{3}}$

$\sf \dfrac{1}{3\sqrt{3}}=\dfrac{Cot \ m - \sqrt{3}}{ 1 + \sqrt{3} Cot \ m}$

$\sf { 1 + \sqrt{3} Cot \ m} =(Cot \ m - \sqrt{3})(3\sqrt{3})$

$\sf { 1 + \sqrt{3} Cot \ m} =3\sqrt{3}Cot \ m - 9$

$\sf 10 =3\sqrt{3}Cot \ m - \sqrt{3} Cot \ m$

$\sf 10 =2\sqrt{3}Cot \ m$

$\sf 5 =\sqrt{3}Cot \ m$

$\sf Cot \ m = \dfrac{5}{ \sqrt{3} }$

$\boxed{ \bold{\large{ \gray{Cot(x - y) = \dfrac{Cot \ x \ Cot \ y + 1}{Cot \ x - Cot \ y}}}}}$

$\sf Cot(60^{\circ} - m) = \dfrac{Cot \ 60^{\circ} Cot \ m + 1}{Cot \ 60^{\circ} - Cot \ m}$

$\sf Cot(60^{\circ} - m) = \dfrac{ \left(\dfrac{1}{ \sqrt{3} } \times \dfrac{5}{ \sqrt{3} } \right) + 1}{\left( \dfrac{1}{ \sqrt{3} } - \dfrac{5}{ \sqrt{3} } \right)}$

$\sf Cot(60^{\circ} - m) = \dfrac{ \dfrac{5}{ 3} + 1}{\left( \dfrac{ - 4}{ \sqrt{3} } \right)}$

$\sf Cot(60^{\circ} - m) = \dfrac{ \dfrac{5 + 3}{ 3}}{\left( \dfrac{ - 4}{ \sqrt{3} } \right)}$

$\sf Cot(60^{\circ} - m) = \dfrac{\dfrac{5 + 3}{ \sqrt{3} }}{ - 4}$

$\sf Cot(60^{\circ} - m) = \dfrac{\dfrac{8}{ \sqrt{3} }}{ - 4}$

$\sf Cot(60^{\circ} - m) = \dfrac{ - 2}{\sqrt{3} }$

VERIFICATION:

$\sf Cot(60^{\circ} + m) = \dfrac{Cot \ 60^{\circ} Cot \ m - 1}{Cot \ 60^{\circ} + Cot \ m}$

$\sf Cot(60^{\circ} + m) = \dfrac{ \left(\dfrac{1}{ \sqrt{3} } \times \dfrac{5}{ \sqrt{3} } \right)- 1}{\left( \dfrac{1}{ \sqrt{3} } + \dfrac{5}{ \sqrt{3} } \right)}$

$\sf Cot(60^{\circ} + m) = \dfrac{ \dfrac{5}{ 3} - 1}{\left( \dfrac{6}{ \sqrt{3} } \right)}$

$\sf Cot(60^{\circ} + m) = \dfrac{ \dfrac{5 - 3}{ 3}}{\left( \dfrac{6}{ \sqrt{3} } \right)}$

$\sf Cot(60^{\circ} + m) = \dfrac{ \dfrac{2}{ \sqrt{3} }}{6}$

$\sf Cot(60^{\circ} + m) = \dfrac{2}{ 6 \sqrt{3} }$

$\sf Cot(60^{\circ} + m) = \dfrac{1}{ 3 \sqrt{3} }$

HENCE VERIFIED.