Question

How do you solve ${\cos ^2}(3x) = 1?$

Answer 1

Hint:In order to solve this trigonometric question, you should have knowledge about the general solution for $\cos \theta = \pm 1$ , in this type of problems you always have to solve for the general solution of the given trigonometric equation.
General solution for $\cos \theta = \pm 1$ is given as $\theta = k\pi ,\;{\text{where}}\;k \in I$

Complete step by step solution:
To solve for ${\cos ^2}(3x) = 1$ we should first consider the argument of the cosine function to be $\theta $ in order to make the process easy and understandable.
$ \Rightarrow \theta = 3x$
So we can rewrite the given trigonometric equation as follows
$
\Rightarrow {\cos ^2}(3x) = 1 \\
\Rightarrow {\cos ^2}\theta = 1 \\
$
Solving it further we will get,
$
\Rightarrow {\cos ^2}\theta = 1 \\
\Rightarrow \cos \theta = \pm 1 \\
$
Now we are all familiar with the general solution of $\cos \theta = 1\;{\text{and}}\;\cos \theta = - 1$,
If you are not then let us understand first what is a general solution in trigonometry.
Since all the trigonometric functions are periodic in nature, that is all of them repeat their values after a fixed interval of angles or you say argument.
So they will definitely have an infinite number of solutions for a particular value. Here the general solution comes: it is the complete set of values of the unknown arguments or angles satisfying the equation.
Now the general solution for $\cos \theta = 1\;{\text{and}}\;\cos \theta = - 1$ are respectively
$\theta = 2k\pi \;and\;\theta = (2k + 1)\pi ,\;{\text{where}}\;k \in I$
From this, we can write the general solution for $\cos \theta = \pm 1$ as
$\theta = k\pi ,\;{\text{where}}\;k \in I$
Therefore we can solve further as
$
\Rightarrow \cos \theta = \pm 1 \\
\Rightarrow \theta = k\pi ,\;{\text{where}}\;k \in I \\
$
Putting back $\theta = 3x$ we will get
$
\Rightarrow \theta = k\pi ,\;{\text{where}}\;k \in I \\
\Rightarrow 3x = k\pi ,\;{\text{where}}\;k \in I \\
\Rightarrow x = \dfrac{{k\pi }}{3},\;{\text{where}}\;k \in I \\
$
Therefore the general solution for $\cos \theta = \pm 1$ is $x = \dfrac{{k\pi }}{3},\;{\text{where}}\;k \in I$

Note: There are three types of solution:
1. Principal solution: It is the smallest value of the unknown angle satisfying the equation.
2. Particular solution: A specific value satisfying the equation.
3. General solution: It is a complete set of values of unknown angles satisfying the equation.
Normally the general solution is preferred.