Question

If the trigonometric function ${\text{tan}}\theta $ can be given as ${\text{tan}}\theta {\text{ = cot}}\left( {\theta + 30^\circ } \right)$, then find the value of the angle θ.

Answer 1

Hint: To find the value of theta denoted by θ, we convert the cotangent function in terms of tangent function and compare their angles to form a relation. We solve the relation for θ.

Complete Step-by-Step solution:
Given Data,
${\text{tan}}\theta {\text{ = cot}}\left( {\theta + 30^\circ } \right)$
We know that,
${\text{tan}}\left( {90^\circ - \theta } \right) = {\text{cot}}\theta $ -- (from the graph of tangent function)
Using this relation we express cot function in terms of tan function to simplify the given relation.
⟹${\text{tan}}\theta $ = ${\text{tan}}\left( {90^\circ - \left( {\theta + 30^\circ } \right)} \right)$
⟹${\text{tan}}\theta $ = ${\text{tan}}\left( {90^\circ - \theta - 30^\circ } \right)$
⟹${\text{tan}}\theta $ = ${\text{tan}}\left( {60^\circ - \theta } \right)$
As the function that is bounded to the angles on both sides is ‘tan’, it is same on both sides so we eliminate them and compare the angles inside the function
⟹θ = 60° - θ
⟹2θ = 60°
⟹θ = 30°

Note: In order to solve this type of question the key is to transform one trigonometric function into another using trigonometric identities. Then we compare the angles and solve the equation to find the answer. Basic knowledge of trigonometric formulae comes in very handy.