Question

If tan A=2/3,then find all the other trigonometric ratios​

Answer 1

Answer:

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Answer 2

Given :-

• Tan A = $\cfrac{2}{3}$

To find :-

• Other Trigonometric Ratios.

Solution :-

We know that:-

$\begin{gathered}\footnotesize{\boxed{\begin{array}{c|c} \bf{\underline{ \: \:Ratio \: \: }} & \bf{\underline{ \: FORMULA \: }} \\ \\ \sf{ \sin(A) }&{ \sf{ \cfrac{ Perpendicular}{ Hypotenuse} }}\\ \\ \sf{ \cos(A) } & { \sf{\cfrac{Base }{ Hypotenuse} }} \\ \\ \sf{ \tan(A) } & \sf \cfrac{ Perpendicular }{Base } \\ \\ \sf{ \cosec(A) } & \sf\cfrac{ Hypotenuse }{ Perpendicular } \\ \\ \sf{ \sec(A) } & \sf\cfrac{ Hypotenuse}{Base} \\ \\ \sf{ \cot(A) } & \sf \cfrac{Base}{Perpendicular} \: \\ \\ \end{array}}}\end{gathered}$

● We have

${\sf{\implies{ \tan A = \cfrac{ Perpendicular}{ Base} = \cfrac{2}{3} }}}$

• Perpendicular = 2
• Base = 3
• Hypotenuse = ?

● So, Hypotenuse can be find by Pythagoras Theorem:-

${\sf{\rightarrow{(Hypotenuse)^2 = (Base)^2 + (Perpendicular)^2}}}$

${\sf{\rightarrow{(Hypotenuse)^2 = (3)^2 + (2)^2}}}$

${\sf{\rightarrow{(Hypotenuse)^2 = 9+4}}}$

${\sf{\rightarrow{(Hypotenuse)^2 = 13}}}$

${\sf{\rightarrow{ \underline{Hypotenuse = \sqrt{13} }}}}$

● So, all other Trigonometric Ratios are :-

${\sf{\implies{\sin A = \cfrac{ Perpendicular }{Hypotenuse}}}}$

${ \boxed{ \red{\sf{\implies{\sin A = \cfrac{ 2}{ \sqrt{13} }}}}}}$

________________

${ \sf{ \implies{ \cos A} = {\cfrac{Base }{ Hypotenuse}}}}$

${ \boxed{ \red{ \sf{ \implies{ \cos A} = {\cfrac{3 }{ \sqrt{13} }}}}}}$

________________

${ \implies{\sf{ \tan A} = \cfrac{ Perpendicular }{Base } }}$

${ \boxed{ \red{ \implies{\sf{ \tan A} = \cfrac{ 2 }{3 } }}}}$

________________

${ \implies\sf{ \cosec A} = \cfrac{ Hypotenuse }{ Perpendicular }}$

${ \boxed{ \red{ \implies\sf{ \cosec A} = \cfrac{ \sqrt{13} }{ 2 }}}}$

________________

${ \implies\sf{ \sec(A) } = \cfrac{ Hypotenuse}{Base} }$

${ \boxed{ \red{ \implies\sf{ \sec(A) } = \cfrac{ \sqrt{13} }{3} }}}$

________________

$\implies \sf{ \cot(A) } = \cfrac{Base}{Perpendicular}$

${ \boxed{ \red{ \implies \sf{ \cot(A) } = \cfrac{3}{2} }}}$