Problem 39 How many cycles does each sine f... [FREE SOLUTION]

Chapter 13: Problem 39

How many cycles does each sine function have in the interval from 0 to 2$\pi ?$ Find the amplitude and period of each function. $$ y=-5 \sin 2 \pi \theta $$

Short Answer

Expert verified

The function $y=-5 \sin 2\pi\theta$ has an amplitude of 5 and a period of $2\pi$ (this implies it completes 1 cycle in the interval from 0 to $2\pi$)

Step by step solution

Identify the Amplitude

From the equation $y=-5 \sin 2\pi\theta$, we see the coefficient before the sine function is -5. The amplitude of a sine function is the absolute value of this coefficient. Therefore the amplitude is |-5| = 5.

Identify the Frequency

In $y=-5 \sin 2\pi\theta$, the coefficient of $\theta$ inside the sine function is $2\pi$. Therefore, the frequency of the sine function is simply that coefficient divided by $2\pi$ so that is $B / 2\pi = 2\pi / 2\pi = 1$.

Calculate the number of cycles

Since the function completes 1 cycle in the interval from 0 to $2\pi$, and we are looking for the numbers of cycles in the interval from 0 to $2\pi$, this sine function completes 1 cycle in that interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude

Amplitude refers to the height of the wave in a trigonometric function, in this case, a sine function. It tells us how far the wave peaks or troughs are from the central axis (midline). Consider the function provided:

In the equation $y = -5 \sin 2\pi\theta$, the number in front of the sine is $-5$.
The amplitude is the absolute value of this coefficient, which is $\left|-5\right| = 5$.

This means that the graph of the sine function will reach up to 5 units above and 5 units below its central axis (midline). Despite the negative sign, amplitude is always expressed as a positive value because we're measuring distance. This negative indicates a vertical flip, not a change in amplitude.

Period

The period of a trigonometric function indicates how long it takes for the function to complete one full cycle, or repetition, of the wave. For the basic sine function $y = \sin \theta$, the period is $2\pi$. This is because it takes $2\pi$ radians to go through all values once, from 0, up and down and back to 0.

In our exercise's function $y = -5 \sin 2\pi\theta$, the period can be calculated as $\frac{2\pi}{B}$, where $B = 2\pi$.
Substituting that value, we get: $\frac{2\pi}{2\pi} = 1$.

Thus, the function completes one full wave cycle in just 1 unit of the $\theta$-axis, rather than the standard $2\pi$ radians.

Frequency

The frequency tells us how many cycles of the sine wave occur in a specific interval, usually measured over $2\pi$ radians. Unlike the period, which tells us how long one cycle takes, frequency indicates how often these cycles occur.

In the equation $y = -5 \sin 2\pi\theta$, the frequency is the coefficient in front of $\theta$, which is $2\pi$.
To find the frequency as it is typically defined over a $2\pi$ interval, we divide this coefficient by $2\pi$, which gives us $\frac{2\pi}{2\pi} = 1$.

This means the frequency is 1, indicating that one cycle happens over the interval from 0 to $2\pi$. Thus, in this context, frequency and period are closely related concepts.

Sine Function

The sine function, represented as $y = \sin \theta$, is fundamental in trigonometry and describes a smooth, periodic oscillation. It is important in modeling various cyclical phenomena, such as sound waves, tides, and light waves. The sine wave begins at zero, reaches a peak, descends back through zero to a trough, and returns back to zero. This full sequence is called a cycle.

It is characterized by a consistent wave pattern that repeats every $2\pi$ radians.
The negative sign in the given function $y = -5 \sin 2\pi\theta$ flips the wave vertically.

Understanding the sine function helps in analyzing real-world phenomena that repeat in regular intervals, making it a powerful tool in both mathematics and physics.

Cycle

A cycle of a sine wave represents one full wave repetition. In terms of the sine function, this means starting from zero, going to a maximum, then descending back through zero to a minimum, and returning to zero.

In the exercise's function $y = -5 \sin 2\pi\theta$, a full cycle encompasses the interval from 0 to 2$\pi$, which is one complete round of the wave.
Normally, a sine wave completes one cycle in an interval of $2\pi$ radians.
However, due to the frequency adjustment $2\pi$ in the provided function, each unit on the $\theta$-axis corresponds to a full cycle.

Recognizing cycles is essential for understanding periodic phenomena, as each cycle represents a repetition of specific patterns or behaviors.

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91影视

How many cycles does each sine function have in the interval from 0 to 2\(\pi ?\) Find the amplitude and period of each function. $$ y=-5 \sin 2 \pi \theta $$

Short Answer

Step by step solution

Identify the Amplitude

Identify the Frequency

Calculate the number of cycles

Key Concepts

Amplitude

Period

Frequency

Sine Function

Cycle

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