Question

How do you solve $5\tan 3x-5=0$ and find all solutions in the interval $\left[ 0,2\pi \right)$?

Answer 1

We will simplify the given expression by keeping the tangent function on one side and the other terms on the other side. Then we will look at the definition of inverse trigonometric function. We will use this definition to find the value of the angle. Then we will get an equation with one variable. We will solve this to obtain the value of $x$.

Complete answer:
The given equation is $5\tan 3x-5=0$. Let us rearrange the terms in the equation so that we have tangent function on one side and the other terms on the other side. So, we have the following equation,
$5\tan 3x=5$
Dividing the above equation by 5, we get
$\tan 3x=1$
Let us look at the definition of inverse trigonometric function. The inverse trigonometric function is the inverse function of a trigonometric function. We use these functions to obtain the value of the angle from any given trigonometric ratio. Applying the inverse trigonometric function for the tangent function, we get the following,
$3x={{\tan }^{-1}}\left( 1 \right)$
We know that the angle which gives the value 1 for tangent function is $\dfrac{\pi }{4}$. Therefore, we have,
$3x=\dfrac{\pi }{4}$
In the interval $\left[ 0,2\pi \right)$, the value of the tangent function will repeat after $\pi$ radians. This is because the period of the tangent function is $\pi$ radians.
Therefore, we also have
$3x=\dfrac{\pi }{4}+\pi$
Solving the above equations for $x$, we get the solution as
$x=\dfrac{\pi }{12}$ and $x=\dfrac{\pi }{12}+\dfrac{\pi }{3}=\dfrac{5\pi }{12}$.

Note: It is important to understand the trigonometric functions and the concept of the inverse trigonometric functions. We should be familiar with the values of the trigonometric functions for standard angles. It is necessary that we are aware of the periodicity of the trigonometric functions. This concept helps us to find the value of the given function within the period of that function.