Problem 31 - Precalculus

Question

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi),\) solve \(f(x)=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 0, \pi, 2\pi\).

Step 1: Identify the Problem

We are given the function \(f(x) = \sin x\) and need to solve the equation \(\sin x = 0\) for \(x\) in the interval \([0, 2\pi)\).

Step 2: Find the Zeros of the Sine Function

The sine function is zero at integer multiples of \(\pi\). This means \(\sin x = 0\) when \(x = n\pi\), where \(n\) is an integer.

Step 3: Consider the Interval Constraint

We need to find values of \(x = n\pi\) in the interval \([0, 2\pi)\). The integer values of \(n\) that satisfy this are \(n = 0, 1, 2\).

Step 4: List the Solutions within the Interval

Thus, the values of \(x\) that satisfy \(\sin x = 0\) within \([0, 2\pi)\) are \(x = 0\), \(x = \pi\), and \(x = 2\pi\).

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is a periodic function, meaning it repeats its values in regular intervals or periods. At the heart of the sine function is its ability to measure the vertical position of a point on the unit circle.
This is done relative to the angle formed with the origin, known as the reference angle. The sine function is given by

• \(f(x) = \sin(x)\)

The function is periodic with a period of \(2\pi\), which indicates that its values repeat every \(2\pi\) radians.
The range of the sine function is

• From -1 to 1, inclusive.

This means no matter the input angle \(x\), the output or function value will always lie within these bounds.
These characteristics make the sine function invaluable for modeling wave-like phenomena, such as sound and light waves.

Interval Notation

Interval notation is a mathematical notation used to represent a set of numbers lying between, or between and including, given endpoints. It's concise and helps in delineating the subset of values we are interested in when solving equations like trigonometric ones.
The interval

• \([0, 2\pi)\)

is an example where square brackets "[ ]" denote inclusion of the endpoint, and parentheses "(" denote exclusion.
Specifically, the square bracket around 0 indicates that 0 is included in the set, and the parenthesis around \(2\pi\) indicates \(2\pi\) is not included.
This means the interval describes all values starting at 0 and up to, but not including, \(2\pi\). Using interval notation provides clarity about which values are being considered in our solution, which is particularly helpful when dealing with periodic functions whose outputs recur over specific intervals.

Zeros of Trigonometric Functions

Finding zeros of trigonometric functions is crucial for solving equations involving them, as zeros mark the points where the function's output is zero. For the sine function, its zeros are integral to understanding its behavior.
The zeros of the sine function are points where the vertical position is zero on the unit circle, corresponding to angles where the sine graph crosses the x-axis.
Mathematically, the sine function is zero at every integer multiple of \(\pi\), represented as:

• \(x = n\pi\)

where \(n\) is an integer.
In practical exercises, like solving \(\sin(x) = 0\) within a specific interval, you must find all integer values of \(x = n\pi\) that fall within the given range.
For

• The interval \([0, 2\pi)\),

the solutions are \(x = 0, \pi, 2\pi\).
Identifying these zeros allows for accurate solutions and provides insight into the periodic nature of trigonometric functions.