Problem 65 - Algebra and Trigonometry

Question

Exercises \(64-66\) will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator. $$ y=4 \sin \left(2 x-\frac{2 \pi}{3}\right) $$ $$ \begin{array}{|c|c|c|c|c|c|} \hline x & {\frac{\pi}{3}} & {\frac{7 \pi}{12}} & {\frac{5 \pi}{6}} & {\frac{13 \pi}{12}} & {\frac{4 \pi}{3}} \\ \hline y & {} & {} & {} \\ \hline \end{array} $$ After completing this table of coordinates, plot the nine ordered pairs as points in a rectangular coordinate system. Then connect the points with a smooth curve.

Step-by-Step Solution

Answer

Begin by calculating the y-values for each x in the equation. Fill the table with these calculations. Plot the resulting points on a graph and connect the dots with a smooth curve following the form of a sine wave.

Step 1: Calculate y-values

Substitute each x-value into the equation, and calculate the corresponding y-value. The equation is in the form of \( y = a \sin(bx - c) \), where \(a = 4\), \(b = 2\), and \(c = \frac{2\pi}{3}\). Therefore we substitute the x-values one by one and calculate the corresponding y-values.

Step 2: Fill the Table

Once the y-values are calculated, fill those values into the table. The Second row of the table is for y-values.

Step 3: Plot Points

The table will give us a set of coordinates (x, y) where x is from the first row and y is from the second row. Then, we can plot these points in a rectangular coordinate system.

Step 4: Connect the Points

Once we have plotted all the points, connect them with a smooth curve to form the sinusoidal graph.

Key Concepts

Understanding the Sine Function

The sine function is a fundamental concept in trigonometry. It represents a periodic wave pattern, which is often used to model oscillating signals. The general form of the sine function is represented as \( y = a \sin(bx - c) \), where:

• \(a\) determines the amplitude, or the height of the peaks and the depth of the troughs.
• \(b\) affects the period or how often the wave cycles over a given interval.
• \(c\) accounts for the horizontal shift or phase shift of the wave.

These parameters influence the shape and position of the sine wave on a graph. This characteristic makes the sine function essential for depicting real-world periodic phenomena such as sound and light waves.

The Coordinate System

A coordinate system is essential for graphing functions like the sine function. It consists of two perpendicular axes—the horizontal axis or x-axis, and the vertical axis or y-axis. In this system:

• Coordinate pairs \((x, y)\) define specific points on the plane.
• This system allows us to visualize mathematical relationships and functions efficiently.

For trigonometric functions, plotting them in a coordinate system helps us see how they behave over different values of \(x\), showcasing their oscillation directly on the graph.

Visualizing the Sinusoidal Graph

A sinusoidal graph is the graphical representation of a sine function. This type of graph exhibits a characteristic wave-like pattern. To construct it from an equation, we:

• Identify key points by substituting specific values of \(x\) to calculate \(y\) values.
• Plot these points accurately on the coordinate system.
• Connect these points smoothly to portray a wave-like pattern.

This graphical representation helps in predicting values and understanding the periodic nature of trigonometric functions, which is crucial in fields like physics and engineering.

Exploring Amplitude

The amplitude of a sine wave measures the wave's height from its central axis to its peak. In the function \( y = a \sin(bx - c) \), the amplitude is given by \( |a| \). It tells us how "tall" each wave is:

• An increase in \(a\) stretches the wave vertically, making it taller.
• A decrease in \(a\) compresses the wave, making it shorter.
• If \(a\) is negative, it flips the wave upside down.

Amplitude influences how strong or intense the wave appears, playing a vital role in applications like audio signal processing where volume correlates with amplitude.

Explaining Phase Shift

Phase shift involves horizontal displacement of the wave, determined by the \(c\) parameter in \( y = a \sin(bx - c) \). This parameter directly affects where the wave starts its cycle on the x-axis. A positive phase shift moves the wave to the right, whereas a negative phase shift moves it to the left. Key features include:

• The formula for the phase shift is \( \frac{c}{b} \).
• A whole wave cycle consists of 360° or \(2\pi\) radians. The phase shift adjusts this starting position.

Understanding phase shift is crucial for synchronizing waves in technological applications, such as radio transmissions, where signals must align precisely.