Question

If \[tan{\text{ }}2A{\text{ }} = {\text{ }}cot\left( {A{\text{ }} - {\text{ }}18^\circ } \right)\], where 2A is an acute angle, find the value of A.

Answer 1

Hint: The main goal in dealing with trigonometric expressions is to simplify them. This means large, multiple-function expressions are considered simplified when they are compact and contain fewer trigonometric functions. Here we already have an expression in tan and cot whereas we need to find a solution of angle A. So we will try to find the value of sin and cos from the value of tan 2A in terms of cot and then substitute the value of found and A-18 in the expression and find the value of A.

Complete step-by-step answer:
2A is an acute angle and therefore lies in the first quadrant.
So, A will be an acute angle and therefore, \[\left( {A{\text{ }} - {\text{ }}18^\circ } \right)\] will also be an acute angle.
So, which implies A and \[\left( {A{\text{ }} - {\text{ }}18^\circ } \right)\] are acute angles.
Because 2A and \[\left( {A{\text{ }} - {\text{ }}18^\circ } \right)\] are acute angles, \[tan{\text{ }}2A\] and \[\cot \left( {A{\text{ }} - {\text{ }}18^\circ } \right)\] will be positive angles as they are in the First Quadrant.
Trigonometric ratio of complementary angles: \[tan{\text{ }}\theta {\text{ }} = {\text{ }}cot\left( {90^\circ {\text{ }} - {\text{ }}\theta } \right)\]
\[ \Rightarrow tan{\text{ 2A }} = {\text{ }}cot\left( {90^\circ {\text{ }} - {\text{ 2A}}} \right)\]
i.e., \[cot{\text{ }}\left( {90{\text{ }} - {\text{ }}2A} \right){\text{ }} = {\text{ }}cot{\text{ }}\left( {A{\text{ }} - {\text{ }}18^\circ } \right)\]
(As already given in the question, \[tan{\text{ }}2A{\text{ }} = {\text{ }}cot\left( {A{\text{ }} - {\text{ }}18^\circ } \right)\])
So, by comparing the angles in the above equation we get
\[\begin{array}{*{20}{l}}
  {\; \Rightarrow 90^\circ - {\text{ }}2A{\text{ }} = {\text{ }}A{\text{ }} - {\text{ }}18^\circ } \\
  { \Rightarrow 108^\circ {\text{ }} = {\text{ }}3A} \\
  { \Rightarrow \angle A{\text{ }} = {\text{ }}36^\circ }
\end{array}\]

The value of angle A is equal to $36^\circ$.

Note: All trigonometric functions are positive in the first quadrant.
Sin and Cosec are positive in the second quadrant.
Tan and Cot are positive in the third quadrant. Cos and Sec are positive in the fourth quadrant.