To find a positive angle less than [tex]\(2\pi\)[/tex] that is coterminal with the given angle [tex]\(\frac{36\pi}{5}\)[/tex], follow these steps:
1. Understand Coterminal Angles:
Coterminal angles are angles that share the same terminal side when drawn in standard position. They can be found by adding or subtracting full circles (multiples of [tex]\(2\pi\)[/tex]).
2. Express the Given Angle:
We start with the given angle:
[tex]\[
\frac{36\pi}{5}
\][/tex]
3. Determine the Reference Full Circle:
Since [tex]\(2\pi\)[/tex] is the measure of one full circle in radians, angles coterminal with [tex]\(\frac{36\pi}{5}\)[/tex] can be found by subtracting multiples of [tex]\(2\pi\)[/tex]:
[tex]\[
k \cdot 2\pi \quad \text{where} \quad k \in \mathbb{Z}
\][/tex]
4. Subtract Multiples of [tex]\(2\pi\)[/tex]:
To get an angle that lies between [tex]\(0\)[/tex] and [tex]\(2\pi\)[/tex], calculate:
[tex]\[
\frac{36\pi}{5} - k \cdot 2\pi
\][/tex]
We need this result to be positive and less than [tex]\(2\pi\)[/tex].
5. Use Modular Arithmetic:
Our goal is to reduce the angle using modulus [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{36\pi}{5} \mod 2\pi
\][/tex]
6. Simplify Calculation:
Given the calculation:
[tex]\[
\frac{36\pi}{5} \mod 2\pi
\][/tex]
In simplified terms, find the remainder when [tex]\(\frac{36\pi}{5}\)[/tex] is divided by [tex]\(2\pi\)[/tex].
7. Result:
The positive angle less than [tex]\(2\pi\)[/tex] that is coterminal with [tex]\(\frac{36\pi}{5}\)[/tex] is:
[tex]\[
3.7699111843077517
\][/tex]
Thus, the positive angle less than [tex]\(2\pi\)[/tex] that is coterminal with [tex]\(\frac{36\pi}{5}\)[/tex] is approximately [tex]\(3.7699\)[/tex] radians.
