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Chapter 8: Problem 34

Solve each equation on the interval \(\sin \left(3 \theta+\frac{\pi}{18}\right)=1\)

Short Answer

Expert verified
Solve for \(sin\left(3x+\frac{pi}{18}\right)=1\theta = \frac{4\pi}{27} + \frac{2k\pi}{3}.\

Step by step solution

01

- Understand Problem Context

We need to solve the equation \(\sin \left(3\theta + \frac{\pi}{18}\right) = 1\) within a specified interval.
02

- Determine the General Solution

Recall that \(\sin x=1\) at \(\frac{\pi}{2} + 2k\pi \) where \((k \in \mathbb{Z})\). Equate \(3\theta +\frac{\pi}{18}\) to \(\frac{\pi}{2} + 2k\pi\): \[\ 3\theta + \frac{\pi}{18} = \frac{\pi}{2} + 2k\pi \]
03

- Isolate \(\theta\)

Solve for \(\theta\) by isolating it on one side of the equation: \[3\theta =\frac{\pi}{2} + 2k\pi - \frac{\pi}{18}\] \[3\theta =\frac{9\pi}{18} + 2k\pi - \frac{\pi}{18}\] \[3\theta = \frac{8\pi}{18} + 2k\pi\] \[\theta = \frac{4\pi}{27} + \frac{2k\pi}{3}\]
04

- Determine Interval Solutions

Now place \(\theta\) back into the interval by changing values for \(k\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

solving trigonometric equations
Solving trigonometric equations often involves finding all angles that satisfy the given equation. In our example, we need to find values of \(\theta\) that make \(\text{sin}(3\theta+\frac{\text{蟺}}{18}) = 1\). This involves several steps: analyzing the sine function, isolating \(\theta\), and ensuring our solutions fit within the given interval. Breaking down each part can make it easier to understand.
sine function
The sine function, denoted as \(\text{sin}(x)\), represents the y-coordinate of a point on the unit circle. Its values range between -1 and 1. Specifically, \(\text{sin}(x)=1\) when the angle \(\text{x}\) is at \(\frac{\text{蟺}}{2}\), added by integer multiples of \(\text{2蟺}\): \(\frac{\text{蟺}}{2}+2k蟺\) where k is any integer. This fundamental property helps us solve the given trigonometric equation.
general solutions in trigonometry
The general solution in trigonometry refers to the formula that covers all possible solutions for a trigonometric equation over the entire set of angles. For the equation \(\text{sin}(3\theta+\frac{\text{蟺}}{18}) = 1\), the general solution involves solving \(\text{3胃}+\frac{\text{蟺}}{18} = \frac{\text{蟺}}{2} + 2k蟺\). By isolating \(\theta\), we get \(\theta = \frac{4\text{蟺}}{27} + \frac{2k蟺}{3}\) which provides a way to find all solutions by varying k.
interval solutions
Interval solutions are specific instances of general solutions that fall within a given range. This means we must adjust k to ensure \(\theta\) lies within the desired interval. If our interval is \([0,2蟺)\), we substitute different values for k into \(\theta = \frac{4蟺}{27} + \frac{2k蟺}{3}\) until all solutions are within this range.
This step refines the general solution to meet any specific constraints provided in the problem.

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