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Chapter 1: Problem 42

In Exercises 37-44, find the reference angle \(\theta^{\prime}\), and sketch \(\theta\) and \(\theta^{\prime}\) in standard position. $$ \theta=\frac{11 \pi}{3} $$

Short Answer

Expert verified
The reference angle \( \theta' = \frac{\pi}{3} \) and it is same as \( \theta = \frac{\pi}{3} \) once \( \theta \) was converted into the range of \(0\) to \(2\pi\). Both can be sketched as a counter-clockwise rotation of \( \frac{\pi}{3} \) radians from the positive x-axis

Step by step solution

01

Convert the Angle to Between 0 and \(2\pi\)

First, determine whether the given angle \( \theta = \frac{11 \pi}{3} \) is between 0 and \(2\pi\). If not, subtract or add \(2\pi\) until the angle falls within this range. Note that \( \frac{11 \pi}{3} = 3.67\pi \) which is greater than \( 2\pi \). Thus subtract \(2\pi\) repeatedly until the angle falls within the range of 0 to \(2\pi\). Doing this results in \( \theta = \frac{\pi}{3} \).
02

Find the Reference Angle

The reference angle is the acute angle the terminal side of the angle makes with the x-axis. Since \( \theta = \frac{\pi}{3} \) lies in the first quadrant and the reference angle for any angle in the first quadrant is just that angle, thus \( \theta' = \frac{\pi}{3} \).
03

Sketch the Angles

To represent \( \theta = \frac{\pi}{3} \) and \( \theta' = \frac{\pi}{3} \), one must sketch a unit circle. \( \theta = \frac{\pi}{3} \) is represented as a counter-clockwise rotation of \( \frac{\pi}{3} \) radians from the positive x-axis and \( \theta' = \frac{\pi}{3} \) is the shortest arc connecting \( \theta \) to the x-axis, which is just \( \theta \) for an angle in the first quadrant.

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