Question

If 2cos 2x =1 then x is equal to

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{2\;cos\,2x=1}$

$\underline{\textbf{To find:}}$

$\textsf{Solution of the trigonometric equation}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{2\;cos\,2x=1}$

$\textsf{This can be written as,}$

$\mathsf{cos\,2x=\dfrac{1}{2}}$

$\mathsf{cos\,2x=cos\,\dfrac{\pi}{3}}$

$\implies\mathsf{2x=2n\pi{\pm}\dfrac{\pi}{3}}$

$\implies\mathsf{x=n\pi{\pm}\dfrac{\pi}{6},\;\;\;n{\in}Z}$

$\underline{\textbf{Formula used:}}$

$\boxed{\begin{minipage}{5cm} \mathsf{If\;cos\,\theta=cos\,\alpha,\;then\;\theta=2n\pi{\pm}\alpha}\\\\\mathsf{where\;n{\in}Z} \end{minipage}}$

Answer 2

Concept

Trignometry is a field of mathematics that investigates the connection between triangle side lengths and angles.

Given

$2cos (2x) =1$

Find

Value of x

Solution

We are given with;

$2cos (2x) =1$

Dividing either sides by 2

$\frac{2cos (2x)}{2} =\frac{1}{2} \\cos (2x) = \frac{1}{2}\\cos (2x) = cos\frac{\pi }{3}$       ....... (1)

Now, we know that

$if\ cos\alpha = cos\beta \\Then\ \alpha = 2n\pi \frac{+}{} \beta$  ....... (2)

Using this formula in equation 1

$2x = 2n\pi \frac{+}{} \frac{\pi}{3} \\x = n\pi \frac{+}{} \frac{\pi}{6},\ n\in Z$

Therefore, $x = n\pi \frac{+}{} \frac{\pi}{6},\ n\in Z$