Question

If $\tan 2A=\cot (A-{{18}^{\circ }})$, where $2A$ is an acute angle, then find the value of $A$.

Answer 1

Hint: Since, $2A$ is an acute angle so we can say that $A$ will also be an acute angle. Thus, all the trigonometric equations applied at this angle will be normal and basic. We will have to convert $\tan 2A$ in the form of the $\cot 2A$ directly, to easily equate both sides of the equation.

“Complete step-by-step answer:”
Here, we have the following given equation as
$\Rightarrow \tan 2A=\cot (A-{{18}^{\circ }})...\text{ (1)}$
We have to convert the $\tan 2A$ in the $\cot 2A$ form.
As per question, $2A$ is an acute angle, then we can say that $A$ will also be an acute angle,
And from trigonometric complementary equations, we have
$\Rightarrow \tan \theta =\cot \left( {{90}^{\circ }}-\theta \right)$
Substituting this value of $\tan \theta $ in terms of $\cot \theta $ in equation (1), we get
\[\begin{align}
  & \Rightarrow \tan 2A=\cot \left( {{90}^{\circ }}-2A \right) \\
 & \Rightarrow \tan 2A=\cot \left( A-{{18}^{\circ }} \right) \\
\end{align}\]
From both of the above equations, we get
\[\Rightarrow \cot \left( {{90}^{\circ }}-2A \right)=\cot \left( A-{{18}^{\circ }} \right)...\text{ (2)}\]
Thus, comparing the angles from both of the sides of the equation (2), we get
\[\Rightarrow {{90}^{\circ }}-2A=A-{{18}^{\circ }}\]
Transposing terms from either side of the equation to keep similar variables at same side, we get
\[\begin{align}
  & \Rightarrow {{90}^{\circ }}-2A=A-{{18}^{\circ }} \\
 & \Rightarrow -2A-A=-{{90}^{\circ }}-{{18}^{\circ }} \\
 & \Rightarrow -3A=-{{108}^{\circ }} \\
\end{align}\]
Now, cancelling out the negation from both sides, we get
\[\begin{align}
  & \Rightarrow -3A=-{{108}^{\circ }} \\
 & \Rightarrow 3A={{108}^{\circ }} \\
\end{align}\]
With cross-multiplication in the above equation, we get
\[\begin{align}
  & \Rightarrow 3A={{108}^{\circ }} \\
 & \Rightarrow A=\dfrac{{{108}^{\circ }}}{3} \\
 & \Rightarrow A={{36}^{\circ }} \\
\end{align}\]
Hence, The value of acute angle \[A={{36}^{\circ }}\].

Note: A probable mistake which can be made here is that, while conversion of one of the trigonometric functions to another form, there could be a sign conventional mistake irrespective of the given information about the acute angle.