/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 24 Although units of \(\pi\) are co... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 24

Although units of \(\pi\) are convenient, we must remember that \(\pi\) is only a number. Numbers that are not multiples of \(\pi\) may be used. Plot the indicated graphs by finding the values of \(y\) that correspond to values of \(x\) of 0,1,2,3,4,5,6, and 7 on a calculator. (Remember, the numbers \(0,1,2,\) and so on represent radian measure.) $$y=-30 \sin x$$

Short Answer

Expert verified
Plot the points from the calculations and join them smoothly for the graph of \( y = -30 \sin x \).

Step by step solution

01

Understanding the Problem Context

The problem requires us to plot the graph of the function \( y = -30 \sin x \) using specific values for \( x \). The range of \( x \) values is \( 0, 1, 2, 3, 4, 5, 6, 7 \). Each of these values represents an angle in radians, and we need to calculate the corresponding \( y \) values using the sine function.
02

Calculate \( y \) for \( x = 0 \)

Substitute \( x = 0 \) into the function: \( y = -30 \sin 0 \). Since \( \sin 0 = 0 \), we find that \( y = -30 \times 0 = 0 \).
03

Calculate \( y \) for \( x = 1 \)

Substitute \( x = 1 \) into the function: \( y = -30 \sin 1 \). Use a calculator to find \( \sin 1 \approx 0.841 \), so \( y \approx -30 \times 0.841 = -25.23 \).
04

Calculate \( y \) for \( x = 2 \)

Substitute \( x = 2 \) into the function: \( y = -30 \sin 2 \). Use a calculator to find \( \sin 2 \approx 0.909 \), so \( y \approx -30 \times 0.909 = -27.27 \).
05

Calculate \( y \) for \( x = 3 \)

Substitute \( x = 3 \) into the function: \( y = -30 \sin 3 \). Use a calculator to find \( \sin 3 \approx 0.141 \), so \( y \approx -30 \times 0.141 = -4.23 \).
06

Calculate \( y \) for \( x = 4 \)

Substitute \( x = 4 \) into the function: \( y = -30 \sin 4 \). Use a calculator to find \( \sin 4 \approx -0.757 \), so \( y \approx -30 \times (-0.757) = 22.71 \).
07

Calculate \( y \) for \( x = 5 \)

Substitute \( x = 5 \) into the function: \( y = -30 \sin 5 \). Use a calculator to find \( \sin 5 \approx -0.958 \), so \( y \approx -30 \times (-0.958) = 28.74 \).
08

Calculate \( y \) for \( x = 6 \)

Substitute \( x = 6 \) into the function: \( y = -30 \sin 6 \). Use a calculator to find \( \sin 6 \approx -0.279 \), so \( y \approx -30 \times (-0.279) = 8.37 \).
09

Calculate \( y \) for \( x = 7 \)

Substitute \( x = 7 \) into the function: \( y = -30 \sin 7 \). Use a calculator to find \( \sin 7 \approx 0.657 \), so \( y \approx -30 \times 0.657 = -19.71 \).
10

Sketch the Graph

With the calculated \( y \)-values for each corresponding \( x \)-value, plot the points on a graph with the horizontal axis as \( x \) and the vertical axis as \( y \). Connect the points with a smooth sinusoidal curve to show the variation of \( y = -30 \sin x \).

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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 is a fundamental building block in trigonometry. It is a periodic function that oscillates between -1 and 1. The sine function is defined for all real numbers and is based on a right-angled triangle.
  • In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.
  • The function repeats its pattern every \(2\pi\) radians.
The general form of the sine function can be expressed as: $$y = A \sin(Bx + C) + D$$ where:
  • \(A\) is the amplitude, which affects the height of the wave.
  • \(B\) affects the period, determining how quickly the cycle repeats. \(B = 1\) results in the standard period of \(2\pi\).
  • \(C\) determines the horizontal shift, moving the graph left or right.
  • \(D\) represents the vertical shift, moving the graph up or down.
In the exercise, the sine function is transformed into $$y = -30 \sin x$$ which indicates an amplitude of 30. The negative sign flips the graph upside-down, causing a reflection across the horizontal axis.
radian measure
Radian measure is a way of expressing angles based on the radius of a circle. This unit of measure is crucial for understanding the sine function and other trigonometric functions because it relates directly to the properties of circles.
  • The radian is defined as the angle created when the arc length is equal to the radius of the circle.
  • One full revolution around a circle is \(2\pi\) radians, equivalent to 360 degrees.
  • Since \(2\pi\) radians equals 360 degrees, \(\pi\) radians corresponds to 180 degrees.
Using radians simplifies many mathematical calculations and is often preferred in calculus and higher-level mathematics. In the exercise, values such as 0, 1, 2, etc., are used as radian measures, allowing us to compute the corresponding values for the sine function for these angles to plot the graph.
graphing techniques
Graphing trigonometric functions requires understanding a few essential techniques to accurately represent their behavior and transformations.
  • Begin by identifying the key components such as amplitude, period, phase shift, and vertical shift.
  • Plot the function at specific intervals, using a calculator for precise sine values if needed.
  • For the function $$y = -30\sin x$$, input values like 0, 1, 2, ..., which are radian measures, into the function to get corresponding \(y\) values.
Once calculated, \(y\)-values such as 0, -25.23, -27.27, etc., indicate how the function behaves at each point. After plotting each calculated point, connect them in a smooth, sinusoidal curve, which typically shows an oscillating wave.
  • Pay attention to the negative sign, which reflects the wave across the horizontal axis.
  • Use a consistent scale on the axes to achieve accurate and proportional representation.
The end result is a complete visual representation of the function that helps in understanding the behavior of the sine function under various transformations.

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Most popular questions from this chapter

In Exercises \(47-50,\) graph the given functions. In Exercises 47 and \(48,\) first rewrite the function with a positive angle, and then graph the resulting function. $$y=0.4|\sin 6 x|$$

Sketch the required curves. The sinusoidal electromagnetic wave emitted by an antenna in a cellular phone system has a frequency of \(7.5 \times 10^{9} \mathrm{Hz}\) and an amplitude of \(0.045 \mathrm{V} / \mathrm{m} .\) Find the equation representing the wave if it starts at the origin. Sketch two cycles.

Solve the given problems. Find the function and graph it for a function of the form \(y=2 \sin b x\) that passes through \((\pi / 6,2)\) and for which \(b\) has the smallest possible positive value.

Sketch the appropriate curves. A calculator may be used. The strain \(e\) (dimensionless) on a cable caused by vibration is \(e=0.0080-0.0020 \sin 30 t+0.0040 \cos 10 t,\) where \(t\) is mea- sured in seconds. Sketch two cycles of \(e\) as a function of \(t\)

In Exercises \(37-40\), use the trace feature of a calculator to find the value of a for either \(y=a \sin x\) or \(y=a \cos x,\) whichever is correct. such that the given point is on the graph. The amplitude of each function is \(2.50 .\) Thereby determine the function. (All points are located such that the \(x\) value is between \(-\pi\) and \(\pi .\) ) (0.67,-1.55)
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