Question

How do we find the maximum value of $y = 1 - \cos 4x$?

Answer 1

Hint: In this question we have to find the bigger value of $y = 1 - \cos 4x$. We use trigonometric identities to solve this question such as $\cos 2\theta = 1 - 2{\sin ^2}\theta $. We rearrange the equation and we find the bigger value of $y$.

Complete step by step solution:
We have $y = 1 - \cos 4x$
We know that
$ \Rightarrow $$\cos 2\theta = 1 - 2{\sin ^2}\theta $
$ \Rightarrow 1 - \cos 2\theta = 2{\sin ^2}\theta $
We can write $\cos 4x = \cos 2(2x)$
So, $1 - \cos 2(2x) = 2{\sin ^2}2x$
The maximum value of ${\sin ^2}2x = 1$
Since the value of $\sin \theta $ lies between $ - 1$ and $1$.
So , $y = 2{\sin ^2}2x$
Put the value of ${\sin ^2}2x = 1$. We get
$y = 2$
Therefore, the maximum value of $1 - \cos 4x$ is $ 2$.

Note: We have two trigonometric identities of $\cos 2\theta $ such as $\cos 2\theta = 1 - 2{\sin ^2}\theta $ and $\cos 2\theta = 2{\cos ^2}\theta - 1$. We have to use $\cos 2\theta = 1 - 2{\sin ^2}\theta $ as by rearranging we get the equation which is given in the equation i.e., $y = 1 - \cos 4x$. We have to use identities according to the requirements.