Question

How do you find the exact value of $\dfrac{7\pi }{3}$ and what quadrant does it go in?

Answer 1

Let $\theta$ be any angle. If the angle $\theta$ lies in the interval of $[0,2\pi ]$, then we can comment about the quadrant it lies in. So, when we divide $[0,2\pi ]$ into four parts, we get the four quadrants. If $\theta$ is greater than $2\pi$, then we have to write the angle in the form such that $\theta =2n\pi +\varphi$. Here, $\varphi$ talks about the quadrant in which $\theta$ lies as $2\pi$ represents a complete cycle.

Complete step by step answer:
As per the given question, we have to find the exact value of the angle given as $\dfrac{7\pi }{3}$ and we have to comment about the quadrant in which angle $\dfrac{7\pi }{3}$ lies in.
We know that the interval $[0,2\pi ]$ is divided into four parts to get the four quadrants.
Length of the interval $[0,2\pi ]$ is $2\pi$. So, when $2\pi$ is divided by 4, we get $\dfrac{2\pi }{4}$ which is equal to $\dfrac{\pi }{2}$. Therefore, starting from 0 to $2\pi$, we have four quadrants each of length $\dfrac{\pi }{2}$.
That is, the interval $\left[ 0,\dfrac{\pi }{2} \right]$ is termed as the first quadrant. The interval $\left[ \dfrac{\pi }{2},\pi \right]$ is termed as the second quadrant. The interval $\left[ \pi ,\dfrac{3\pi }{2} \right]$ is termed as third quadrant and the remaining part, that is, the interval $\left[ \dfrac{3\pi }{2},2\pi \right]$ is termed as the fourth quadrant.
If we have an angle greater than $2\pi$, then we express it as $\theta =2n\pi +\varphi$ where $\varphi$ talks about the quadrant in which $\theta$ lies as $2\pi$ represents a complete cycle.

Now, we are given an angle $\dfrac{7\pi }{3}$. We can rewrite the numerator in the form of $3n\pi +\varphi$ for simplification since the denominator is 3. Then, we get
$\Rightarrow \dfrac{7\pi }{3}=\dfrac{6\pi +\pi }{3}=2\pi +\dfrac{\pi }{3}$

As we know $2\pi$ represents a complete cycle, we can write
$\Rightarrow 2\pi +\dfrac{\pi }{3}=\dfrac{\pi }{3}$

Here, we have $\dfrac{\pi }{3}$ which lies in the interval $\left[ 0,\dfrac{\pi }{2} \right]$.
$\therefore \dfrac{\pi }{3}$ is the exact value of $\dfrac{7\pi }{3}$ and it is located in the first quadrant.

Note: While solving such types of problems, we must have enough knowledge about the intervals of four quadrants and the way they are divided. We have to simplify the given angle into a simple one by converting like $\theta =2n\pi +\varphi$, which is simply equal to $\varphi$. We must avoid calculation mistakes while converting the angle.