Question

find a positive angle less than 2π that is coterminal with the given angle.
\frac{33pi}{4}
a positive angle less than 2π that is coterminal with \frac{33pi}{4} 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 \(0<\frac{33\pi}{4}-2k\pi < 2\pi\).

Step2: Solve the inequality for \(k\)

First, set up the left - hand side of the inequality \(\frac{33\pi}{4}-2k\pi>0\), which simplifies to \(\frac{33\pi}{4}>2k\pi\), then \(\frac{33}{4}>2k\), and \(k < \frac{33}{8}=4.125\).
Next, set up the right - hand side of the inequality \(\frac{33\pi}{4}-2k\pi<2\pi\). Rearrange it: \(\frac{33\pi}{4}-2\pi<2k\pi\), \(\frac{33\pi - 8\pi}{4}<2k\pi\), \(\frac{25\pi}{4}<2k\pi\), \(\frac{25}{4}<2k\), \(k>\frac{25}{8} = 3.125\).
Since \(k\) is an integer, \(k = 4\).

Step3: Calculate the coterminal angle

Substitute \(k = 4\) into \(\frac{33\pi}{4}-2k\pi\). We get \(\frac{33\pi}{4}-2\times4\pi=\frac{33\pi}{4}-8\pi=\frac{33\pi - 32\pi}{4}=\frac{\pi}{4}\).

Answer:

\(\frac{\pi}{4}\)