Question

find a positive angle less than 2π that is coterminal with the given angle. - 25π/3 a positive angle less than 2π that is coterminal with - 25π/3 is . (simplify your answer. type your answer in terms of π. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall coterminal - angle formula

Coterminal angles differ by \(2k\pi\), where \(k\) is an integer. We want to find \(k\) such that \(-\frac{25\pi}{3}+2k\pi>0\) and \(-\frac{25\pi}{3}+2k\pi < 2\pi\).

Step2: Solve the inequality for \(k\)

First, set \(-\frac{25\pi}{3}+2k\pi>0\). Then \(2k\pi>\frac{25\pi}{3}\), so \(k > \frac{25}{6}\approx4.17\).
Next, set \(-\frac{25\pi}{3}+2k\pi<2\pi\). Rearrange it to \(2k\pi<2\pi+\frac{25\pi}{3}=\frac{6\pi + 25\pi}{3}=\frac{31\pi}{3}\), then \(k<\frac{31}{6}\approx5.17\).
Since \(k\) is an integer, \(k = 5\).

Step3: Calculate the coterminal angle

Substitute \(k = 5\) into \(-\frac{25\pi}{3}+2k\pi\). We get \(-\frac{25\pi}{3}+2\times5\pi=-\frac{25\pi}{3}+10\pi=-\frac{25\pi}{3}+\frac{30\pi}{3}=\frac{5\pi}{3}\).

Answer:

\(\frac{5\pi}{3}\)