Question

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

Answer 1

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.