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Chapter 11: Problem 3

In \(3-14,\) sketch one cycle of the graph. $$ y=2 \sin x $$

Short Answer

Expert verified
Graph has amplitude of 2 and period of \(2\pi\), oscillating between \([-2, 2]\).

Step by step solution

01

Identify Key Attributes of Sine Function

The sine function, \( y = \sin x \), completes one cycle over the interval from \( 0 \) to \( 2\pi \). The graph oscillates between -1 and 1. The function \( y = 2 \sin x \) is similar but the amplitude is multiplied by 2, so it oscillates between -2 and 2.
02

Determine Key Points in the Cycle

The key points to plot a cycle of the sine function \( y = 2 \sin x \) are \((0, 0)\), \((\frac{\pi}{2}, 2)\), \((\pi, 0)\), \((\frac{3\pi}{2}, -2)\), and \((2\pi, 0)\). These correspond to \(0, \pi / 2, \pi, 3\pi / 2,\) and \ 2\pi\, the integer multiples of \( \pi / 2 \).
03

Sketch the Graph

Start by drawing the x-axis and y-axis. Plot the points determined in the previous step: \((0, 0)\), \((\frac{\pi}{2}, 2)\), \((\pi, 0)\), \((\frac{3\pi}{2}, -2)\), and \((2\pi, 0)\). Draw a smooth, continuous curve through these points to complete one cycle of the sine wave.
04

Verify Amplitude and Period

Ensure that the peak of the graph reaches 2 and the trough reaches -2. The period should be \(2\pi\), which means one complete cycle from \(0\) to \(2\pi\). This confirms that the function's amplitude and period match \( y = 2 \sin x \).

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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 concept in trigonometry that describes a smooth periodic oscillation. It is typically represented as \(y = \sin x\), and it maps an angle \(x\) to a point on the unit circle. This mapping results in a wave-like graph that oscillates between -1 and 1.
The period of the standard sine function is \(2\pi\), meaning it takes an angle of \(2\pi\) to complete one full cycle. This periodic nature is what creates the familiar undulating sine wave when plotted on a graph.
Often referred to as a wave function, the sine function models various real-life periodic phenomena like sound waves, light waves, and even tides.
  • Standard Function: \(y = \sin x\)
  • Period: \(2\pi\)
  • Range: [-1, 1]
Amplitude
Amplitude refers to the height of the wave's peaks from its central axis in a wave-like function. It measures how far the function moves away from the center line where the function crosses zero.
In our particular function \(y = 2 \sin x\), the amplitude is 2, which means the wave oscillates between -2 and 2. This enhanced amplitude is a result of multiplying the sine function by 2, essentially stretching the wave vertically.
Understanding the amplitude is crucial in physics and engineering as it can determine the strength or intensity of a signal or wave.
  • Amplitude Formula: The coefficient in front of \(\sin x\)
  • In \(y = 2 \sin x\), Amplitude: 2
  • Effect: Increased peak and trough distance from 0
Graph Sketching
Graph sketching is a technique used to visually represent mathematical functions on a coordinate plane. It's especially useful for understanding trigonometric functions like the sine function involved in our example.
To sketch \(y = 2 \sin x\), begin by drawing the x-axis and y-axis. Use the key points identified - \((0, 0), (\frac{\pi}{2}, 2), (\pi, 0), (\frac{3\pi}{2}, -2), (2\pi, 0)\) - to guide the wave's plot. Mark these points and ensure your graph passes smoothly through them, forming a sine wave.
  • Identify key points between \(0\) and \(2\pi\).
  • Ensure the peaks and troughs match the amplitude.
  • Draw smoothly through plotted points for accurate representation.
This visualization provides valuable insights into the behavior of trigonometric functions, helping in problems involving waves and oscillations.

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Most popular questions from this chapter

Recall from your geometry course that a polygon is circumscribed about a circle if each side of the polygon is tangent to the circle. Since each side is tangent to the circle, the radius of the circle is perpendicular to each side at the point of tangency. We will use the tangent tunction to examine the formula for the perimeter of a circumscribed regular polygon. Let square \(A B C D\) be circumscribed about circle \(O .\) A radius of the circle, \(\overline{O P},\) is perpendicular to \(\overline{A B}\) at \(P .\) (1) In radians, what is the measure of \(\angle A O B ?\) (2) Let \(m \angle A O P=\theta .\) If \(\theta\) is equal to one-half the measure of \(\angle A O B\) , find \(\theta .\) (3) Write an expression for \(A P\) in terms of \(\tan \theta\) and \(r,\) the radius of the circle. (4) Write an expression for \(A B=s\) in terms of \(\tan \theta\) and \(r\) (5) Use part \((4)\) to write an expression for the perimeter in terms of \(r\) and the number of sides, \(n\) . b. Let regular pentagon \(A B C D E\) be circumscribed about circle \(O .\) Repeat part a using pentagon \(A B C D E .\) c. Do you see a pattern in the formulas for the perimeter of the square and of the pentagon? If so, make a conjecture for the formula for the perimeter of a circumscribed regular polygon in terms of the radius \(r\) and the number of sides \(n .\)

Sketch one cycle of each function. \(y=\sin x\)

Sketch the graph of \(y=\cos x\) in the interval \(0 \leq x \leq 4 \pi\) a. In the interval \(0 \leq x \leq 4 \pi,\) for what values of \(x\) is the graph of \(y=\cos x\) increasing? b. In the interval \(0 \leq x \leq 4 \pi,\) for what values of \(x\) is the graph of \(y=\cos x\) decreasing? c. How many cycles of the graph of \(y=\cos x\) are in the interval \(0 \leq x \leq 4 \pi ?\)

In \(15-26,\) find each exact value in radians, expressing each answer in terms of \(\pi\) \(y=\arccos \frac{1}{2}\)

Find the amplitude of each function. \(y=\frac{1}{8} \sin x\)
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