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Chapter 5: Problem 14

Find the four smallest positive numbers \(\theta\) such that \(\sin \theta=0\).

Short Answer

Expert verified
The four smallest positive values of \(\theta\) that satisfy the equation \(\sin \theta=0\) are \(\theta = \pi, 2\pi, 3\pi, 4\pi\).

Step by step solution

01

Find the sine function zeros

Recall that \(\sin \theta=0\) when \(\theta\) is an integer multiple of \(\pi\). This means that we can represent the zeros of the sine function as \(\theta = n\pi\), where \(n\) is an integer.
02

Find the smallest positive values for \(n\)

Since we are looking for the four smallest positive values of \(\theta\), we want to find the smallest positive integers \(n\) that make \(\theta = n\pi\) true. These integers are \(n = 1, 2, 3, 4\).
03

Substitute the values of \(n\) into the equation

Now that we have found the smallest positive values for \(n\), we can substitute them into the equation \(\theta = n\pi\) to find the corresponding values of \(\theta\). For \(n = 1\): \(\theta = 1\pi = \pi\) For \(n = 2\): \(\theta = 2\pi = 2\pi\) For \(n = 3\): \(\theta = 3\pi = 3\pi\) For \(n = 4\): \(\theta = 4\pi = 4\pi\)
04

State the four smallest positive values of \(\theta\)

The four smallest positive values of \(\theta\) that satisfy the equation \(\sin \theta=0\) are: \(\theta = \pi, 2\pi, 3\pi, 4\pi\)

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