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Chapter 6: Problem 6

Find all solutions on the interval \(0 \leq \theta<2 \pi\). $$ \sin (\theta)=0 $$

Short Answer

Expert verified
The solutions are \(\theta = 0\) and \(\theta = \pi\).

Step by step solution

01

Identify Roots of the Sine Function

The sine function, \(\sin(\theta)\), is equal to zero at specific values of \(\theta\). This occurs whenever \(\theta = n\pi\), where \(n\) is an integer. For the interval \(0 \leq \theta < 2\pi\), we need to identify these specific integer multiples of \(\pi\).
02

Determine Specific Solutions in the Given Interval

Within the interval \(0 \leq \theta < 2\pi\), we need non-negative integer values for \(n\) such that the solutions \(\theta = n\pi\) are valid. This includes \(\theta = 0\), \(\theta = \pi\), and \(\theta = 2\pi\). However, since \(2\pi\) is not less than \(2\pi\), we do not include it as per the interval constraint. Thus, the solutions are \(\theta = 0\) and \(\theta = \pi\).

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Key Concepts

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

Sine Function
The sine function, often written as \( \sin(\theta) \), is a trigonometric function that arises in the study of angles and triangles. At its core, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse in a right triangle. However, beyond triangles, the sine function is fundamental in the world of periodic phenomena, such as sound waves and circular motion. When graphed, the sine function forms a wave-like pattern that repeats every \( 2\pi \). This means every \( 2\pi \), the graph pattern repeats, making sine a periodic function.
  • On its graph, \( \sin(\theta) = 0 \) at angles that are integer multiples of \( \pi \), such as \( \theta = 0, \pi, 2\pi, 3\pi, \ldots \).
  • This is because in the unit circle, these particular points correspond to where the circle intersects the x-axis.
Recognizing these zeroes is key when solving trigonometric equations, especially those set in specific intervals.
Unit Circle
The unit circle is a circle with a radius of exactly one unit centered at the origin of a coordinate plane. It is an essential tool in trigonometry because it illustrates the functions \( \sin\) and \( \cos\) for all angles geometrically. The angle \( \theta \) is measured starting from the positive x-axis, and extends counterclockwise around the circle.
  • On the unit circle, the x-coordinate of a point represents the cosine of that angle, while the y-coordinate represents the sine.
  • Thus, at angles where the sine is zero, the point lies directly on the x-axis.
Understanding the position of angles on the unit circle helps in identifying angle measures that make the sine zero, such as \( 0 \) and \( \pi \), which correlate with the x-axis. It's a visual approach to solving trigonometric equations within specified intervals.
Interval Notation
Interval notation is a way of denoting all numbers between a set of numbers called endpoints. It is particularly useful for expressing solutions to inequalities and equations over a certain range. In trigonometry, interval notation helps define the specific range over which we want to find solutions.
  • For example, the interval \( [0, 2\pi) \) indicates that \( \theta \) should be considered from 0 to just under \( 2\pi \), including 0 but excluding \( 2\pi \).
  • Opening brackets \( ( \) or \( ) \) indicate that the endpoint is not included, while square brackets \( [ \) or \( ] \) mean it is included.
This way of denoting ranges ensures clarity when solving trigonometric equations like \( \sin(\theta) = 0 \) within specific boundaries, making it precise and comprehensive for problems requiring such specifications.

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Most popular questions from this chapter

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