Question

If $\tan 2\theta \tan \theta =1$, then the general value of $\theta $ is [Roorkee 1980; Karnataka CET 1992, 93, 2003]
(A) $(n+\frac{1}{2})\frac{\pi }{3}$
(B) $(n+\frac{1}{2})\pi $
(C) $(2n\pm \frac{1}{2})\frac{\pi }{3}$
(D) None of these

Answer 1

Hint: This is a trigonometric question. The branch of mathematics known as trigonometry studies how triangles' sides and angles correlate with one another. In order to solve the above problem, first we will take $\tan \theta $ to the opposite side and convert it into cot form, followed by $\tan ({{90}^{o}}-\theta )$ form. We will then compare the angles of both to get the value of $\theta $.

Complete step by step Solution:
The given equation is $\tan 2\theta \tan \theta =1$
Solving the equation
$\tan 2\theta =\frac{1}{\tan \theta }$
As cotangent is the inverse of tangent. So,
$\tan 2\theta =\cot \theta $
Now, we know that $\cot \theta $ can also be written as$\tan (\frac{\pi }{2}-\theta )$. Therefore,
$\tan 2\theta =\tan (\frac{\pi }{2}-\theta )$
$2\theta =n\pi +\frac{\pi }{2}-\theta $
$3\theta =n\pi +\frac{\pi }{2}$
$\theta =\frac{n\pi }{3}+\frac{\pi }{6}$
Taking $\frac{\pi }{3}$ as common from both the terms, we get the value of $\theta $ as
$\theta =(n+\frac{1}{2})\frac{\pi }{3}$
Thus, if$\tan 2\theta \tan \theta =1$, then the general value of $\theta $ is $\theta =(n+\frac{1}{2})\frac{\pi }{3}$

Therefore, the correct option is (A).

Additional Information: Trigonometric identities are equivalences in trigonometric functions. The trigonometric ratio is the relationship between the angle measurement and the side length of a right-angle triangle. Trigonometric function has widely used applications. These functions can be used for measuring the height of buildings or mountains. It can also be used in navigation.



Note: There are six different trigonometric functions. These trigonometric functions are: Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent. There are various trigonometric equations showing relationships between these trigonometric functions. The trigonometric questions can be solved easily by knowing the positive or negative signs to use and doing the calculations carefully.