Problem 66 - 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. $$ y=3 \sin \frac{\pi}{2} x $$ $$ \begin{array}{lllllllll} {x} & {0} & {\frac{1}{3}} & {1} & {\frac{5}{3}} & {2} & {\frac{7}{3}} & {3} & {\frac{11}{3}} & {4} \\ {y} & {} & {} & {} & {} & {} & {} & {} \end{array} $$ 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

Following the steps, the complete table of coordinates becomes: {x: 0, y: 0}, {x: 1/3, y: 3}, {x: 1, y: 3}, {x: 5/3, y: -3}, {x: 2, y: 0}, {x: 7/3, y: 3}, {x: 3, y: 3}, {x: 11/3, y: -3}, {x: 4, y: 0}. By plotting these points, a wave curve on the rectangular coordinate system can be drawn to represent the function.

Step 1: Apply the Sine Function

Use the provided sine function \(3sin(\frac{\pi}{2}x)\) and begin substituting each value of x into the equation, from the table, to obtain the corresponding y-values.

Step 2: Calculate y-values for each x

Upon substituting each value of x, we will get the corresponding y-values. The outputs yield the following values: \When \(x = 0\), \(y = 3sin(\frac{\pi}{2}*0) = 0\). \When \(x=\frac{1}{3}\), \(y = 3sin(\frac{\pi}{2}*\frac{1}{3}) = 3\). \When \(x=1\), \(y = 3sin(\frac{\pi}{2}*1) = 3\). \When \(x=\frac{5}{3}\), \(y = 3sin(\frac{\pi}{2}*\frac{5}{3}) = -3\). \When \(x=2\), \(y = 3sin(\frac{\pi}{2}*2) = 0\). \When \(x=\frac{7}{3}\), \(y = 3sin(\frac{\pi}{2}*\frac{7}{3}) = 3\). \When \(x=3\), \(y = 3sin(\frac{\pi}{2}*3) = 3\). \When \(x=\frac{11}{3}\), \(y = 3sin(\frac{\pi}{2}*\frac{11}{3}) = -3\). \When \(x=4\), \(y = 3sin(\frac{\pi}{2}*4) = 0\).

Step 3: Plot Points and Draw Curve

Once the y-values are obtained, we'll plot all the (x,y) coordinate points onto the coordinate system and draw a smooth curve that connects all of these points. The curve reflects the graph of \(y = 3sin(\frac{\pi}{2}x)\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in the study of mathematics, especially when dealing with angles and the geometry of triangles. One of these functions is the sine function, denoted as 'sin'. The sine of an angle in a right-angled triangle is the ratio of the length of the opposite side to the angle to the length of the triangle's hypotenuse. In a broader sense, the sine function is used to describe oscillatory and wave patterns, which is why it is also crucial in physics and engineering.

When graphing trigonometric functions like the sine function, we usually work within a coordinate system where the horizontal axis (x-axis) represents the angle or the time, and the vertical axis (y-axis) represents the value of the sine function at that angle or time. The sine function is periodic, which means it repeats its values in regular intervals, and these intervals are called periods. The basic sine function has a period of \(2\pi\), which shows how the pattern of the sine function repeats every \(2\pi\) units along the x-axis.

Coordinate System

The coordinate system is a two-dimensional framework used for plotting points, lines, and curves. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis. Points in this system are represented by pairs of numbers known as coordinates, which define their location with respect to the origin (0,0), which is where the two axes intersect.

In the context of trigonometry, the coordinate system is used to visualize the behavior of trigonometric functions, such as the sine function. The x-axis typically represents the independent variable (such as the angle in radians, in the case of sine), while the y-axis represents the dependent variable (the function's output value).

Plotting Points

Plotting points is the act of marking a point on the coordinate system based on its coordinates. To plot a point with coordinates \(x, y\), you start at the origin, move \(x\) units along the x-axis, and then move \(y\) units up or down along the y-axis, depending on whether \(y\) is positive or negative. It's like navigating a map: you go right or left for the x-coordinate and up or down for the y-coordinate.

When plotting points for the sine function, you're essentially mapping the output of the function at various angles or points in time. After plotting enough points, you get a visual representation of the function, which you can then connect with a smooth curve to see the overall shape.

Sine Curve Properties

The sine curve has specific properties that consistently define its shape and behavior. Some of these properties include its amplitude, period, phase shift, and vertical shift. The amplitude is the maximum distance the function's graph is from the x-axis, determining the 'height' of the wave. For example, in the function \(y = 3\sin(\frac{\pi}{2}x)\), the amplitude is 3, indicating that the graph reaches 3 units above and below the x-axis at its peaks.

The period is the distance over which the function's graph repeats itself, and it is typically \(2\pi\) for the basic sine function. However, multiplying the variable inside the sine function by a constant, such as \(\frac{\pi}{2}\), affects the period of the graph. Phase shift indicates any horizontal movement of the graph to the left or right, while vertical shift indicates any up or down movement of the graph along the y-axis.

Recognizing these properties is crucial for drawing accurate graphs of sine functions without relying on plotting individual points. They help you understand how changes to the sine function's equation affect the graph's shape and position.