Problem 18 - Precalculus Essentials

Question

Find all solutions of each equation. $$\sin x=0$$

Step-by-Step Solution

Answer

The solution to the equation \( \sin x = 0 \) are all multiples of \( \pi \), i.e., \( x = n \pi \) where \( n \) is an integer.

Step 1: Understand the function

The primary values which you're interested in are within the unit circle, where the sine function varies from -1 to 1.

Step 2: Identify when sin(x) equals 0

The sine function equals zero at two key points within one period: \( x = 0 \) and \( x = \pi \).

Step 3: Extend to all solutions

Because the sine function repeats every \( 2\pi \), you can get all solutions by adding multiples of \( 2\pi \) to these two key points. Therefore, the general solutions are \( x = n \cdot 2\pi \) and \( x = (n + 0.5) \cdot 2\pi \) where \( n \) is an integer. Combining these gives \( x = n\pi \), where \( n \) is an integer.

Key Concepts

Sine Function

The sine function is a fundamental aspect of trigonometry, representing the y-coordinate of a point on the unit circle corresponding to a given angle. It is crucial to understand that the sine of an angle measures the vertical position of a point on this circle, which can range between -1 and 1.
Let's break it down:

• When the angle is 0 or a multiple of \(\pi\), the sine value is 0 because the point is on the horizontal axis.
• At \(\pi/2\), the sine value reaches its maximum of 1, and at \(-\pi/2\), it reaches its minimum of -1.
• The sine function is periodic, meaning it repeats its values at regular intervals.

The periodic nature of the sine function is why solving \(\sin x = 0\) involves finding not just one solution, but potentially many repeated at regular intervals.

Unit Circle

The unit circle is an essential tool in trigonometry. It's a circle with a radius of 1, centered at the origin of a coordinate plane. This circle helps to visualize and understand trigonometric functions by providing a reference for angles and their corresponding sine values.
Some important points to consider:

• The angle \(x\) in radians is measured from the positive x-axis, moving counterclockwise. For negative angles, the direction is clockwise.
• The angles where the sine function is zero correspond to the points \((1, 0)\) and \((-1, 0)\), which occur at 0 and \(\pi\).
• This visual representation makes it easier to see why the sine of an angle might equal zero at these points and how the periodic nature manifests by going around the circle.

Using the unit circle, one can visualize the periodicity of the sine function, identifying the values where the sine function returns to zero as the process repeats.

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. A key characteristic of the sine function is its periodic nature, with a standard period of \(2\pi\). This means that after an angle increases by \(2\pi\), the sine value repeats.
Here's how periodicity applies to the equation \(\sin x = 0\):

• The function returns to zero every \(\pi\) radians because the pattern of the sine wave brings it back to the horizontal axis.
• The general solution for \(\sin x = 0\) can thus be expressed as \(x = n\pi\), where \(n\) is any integer. This accounts for every intersection of the sine wave with the x-axis.
• Periodic functions like sine are foundational in modeling wave patterns, oscillations, and any repetitive phenomena.

Grasping the concept of periodicity allows you to predict and find not just one solution, but infinitely many, repeating in an ordered manner for trigonometric equations.