Question

If $\tan \theta =\cot \left( {{30}^{\circ }}+\theta \right)$, then what is the value of $\theta$ ?

Answer 1

In the given question, we are given an expression based on trigonometric functions. And we are asked in the question to find the value of $\theta$ using the given expression. So, using the conversions of the trigonometric functions such as $\tan \theta =\cot \left( {{90}^{\circ }}-\theta \right)$, we will substitute in the given expression. We will then have the trigonometric functions gone and have the angles across the equality sign. We will then solve the expression further and find the value of $\theta$. Hence, we will have the value of $\theta$ which will be in line or will satisfy the given expression.

Complete step by step solution:
According to the given question, we have an expression given to us which is based on trigonometric functions. We have to solve the given expression and find the value of $\theta$.
The expression given to us is,
$\tan \theta =\cot \left( {{30}^{\circ }}+\theta \right)$----(1)
We can see in the above equation (1), that both the trigonometric functions used across the equality sign are different. So, we will change or convert one of the trigonometric functions into the other form.
We know that, $\tan \theta =\cot \left( {{90}^{\circ }}-\theta \right)$. Applying this in equation (1), we will get the new expression as,
$\Rightarrow \cot \left( {{90}^{\circ }}-\theta \right)=\cot \left( {{30}^{\circ }}+\theta \right)$---(2)
In the equation (2), since the trigonometric functions are same so we can proceed on to equating their angles, so we will get,
$\Rightarrow {{90}^{\circ }}-\theta ={{30}^{\circ }}+\theta$----(3)
In the equation (3), taking the theta terms to one side and the numerical angles to another side, we will have the value of the theta as,
$\Rightarrow 2\theta ={{90}^{\circ }}-{{30}^{\circ }}$
Solving the above expression further, we will have the new expression as,
$\Rightarrow 2\theta ={{60}^{\circ }}$
Now, we will divide 60 degrees by 2 and we will have the value of $\theta$, that is,
$\Rightarrow \theta ={{30}^{\circ }}$
Therefore, the value of $\theta ={{30}^{\circ }}$.

Note: The conversion of the tangent function to cotangent function should be done clearly. Also, the obtained value can be verified if it is correct or not and for that we will substitute $\theta ={{30}^{\circ }}$ in the given expression. We will have the value on substitution as,
$\tan {{30}^{\circ }}=\cot \left( {{30}^{\circ }}+{{30}^{\circ }} \right)$
We get,
$\Rightarrow \tan {{30}^{\circ }}=\cot \left( {{60}^{\circ }} \right)=\dfrac{1}{\sqrt{3}}$
That is, we got the correct value of $\theta ={{30}^{\circ }}$.