Question

How do you solve the following equation $2\cos 3x = 1$ in the interval $\left[ {0,\pi ?} \right]$

Answer 1

Hint: In this question, we are going to solve the given equation for the given interval.
 Solve the equation by using the substitution method and then solving it we get the values of $x$ from the unit circle.
Thus we can get the required result from the given interval.

Complete step-by-step solution:
In this question, we are going to solve the given equation for the given interval.
First write the given equation and mark it as $\left( 1 \right)$
$2\cos 3x = 1...\left( 1 \right)$
Now we are going to use the substitution method,
Let us substitute $3x = t$ in the above equation we get,
$ \Rightarrow 2\cos t = 1$
We can rewrite the above equation as
$ \Rightarrow \cos t = \dfrac{1}{2}$
$ \Rightarrow t = {\cos ^{ - 1}}\left( {\dfrac{1}{2}} \right)$
We get the values of $t$ from the unit circle.
$ \Rightarrow t = \dfrac{\pi }{3},\dfrac{{5\pi }}{3},\dfrac{{7\pi }}{3}$
Substitute back $t = 3x$in the given equation we get
Hence,$3x = 2n\pi \pm \dfrac{\pi }{3}$, where $n$ is an integer.
Hence $3x$ can take the value,
$ \Rightarrow 3x = \dfrac{\pi }{3},3x = \dfrac{{5\pi }}{3},3x = \dfrac{{7\pi }}{3}$
Hence $x$ can take the value,
$ \Rightarrow x = \dfrac{\pi }{{3 \times 3}},x = \dfrac{{5\pi }}{{3 \times 3}},x = \dfrac{{7\pi }}{{3 \times 3}}$
On multiply the term and we get
$ \Rightarrow x = \dfrac{\pi }{9},x = \dfrac{{5\pi }}{9},x = \dfrac{{7\pi }}{9}$
On we get,
$ \Rightarrow x = \dfrac{\pi }{9},\dfrac{{5\pi }}{9},\dfrac{{7\pi }}{9}$

The values that $x$ can take in the interval $\left[ {0,\pi } \right]$ are $\left\{ {\dfrac{\pi }{9},\dfrac{{5\pi }}{9},\dfrac{{7\pi }}{9}} \right\}$.

Note: Solving the trigonometric equation is a tricky work that often leads to errors and mistakes. Therefore, answers should be carefully checked. After solving, you can check the answers by using a graph.
The unit circle or trigonometric circle as it is also known is useful to know because it lets us easily calculate the cosine, sine, and tangent of any angle between $0$ and $360$ degrees.
Trigonometry is also helpful to measure the height of the mountain, to find the distance of long rivers, etc. its applications are in various fields like oceanography, astronomy, navigation, electronics, physical sciences etc.
The steps for solving trigonometric equation:
Put the equation in terms of one function of one angle.
Write the equation as one trigonometric function of an angle equals a constant.
Write down the possible values for the angle.
If necessary solve for the variable