Problem 35 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The function \(y=sin(5θ)\) has 5 cycles in the interval from 0 to \(2π\), an amplitude of 1, and a period of \(2π/5\).

Step 1: Identify the frequency of the function

The frequency of the function is the coefficient of the variable θ, which is 5 in this case.

Step 2: Compute the number of cycles

The number of cycles is equivalent to the frequency for a sine function in the interval from 0 to 2π. So, there are 5 cycles in this interval.

Step 3: Identify the amplitude of the function

The amplitude of a function is the absolute value of the coefficient of the sin function, which means the amplitude is 1.

Step 4: Compute for the period of the function

The period of the function is found by taking \(2π\) divided by the frequency. Hence, the period of the function y = sin(5θ) is \(2π/5\).

Key Concepts

Amplitude

Amplitude refers to the height of the wave from its central axis to its peak peak or trough. In the context of trigonometric functions, and specifically in a sine function like \( y = \sin 5\theta \), the amplitude is determined by the coefficient in front of the sine function itself.

• For \( y = \sin 5\theta \), this coefficient is 1, as the equation does not have an explicit number in front of \( \sin \).
• This tells us that the wave's peak and trough deviate by 1 unit up and 1 unit down from the center line, typically the x-axis when plotted on a graph.

Understanding amplitude helps in visualizing the extent of oscillation of the wave along the vertical axis, which in turn represents how vigorous or gentle the wave's motion is in a variety of physical situations.

Period

The period of a trigonometric function describes how long it takes to complete one full cycle of the wave. It's about repeated motions and patterns. For sine functions, the formula for the period is generally \[\frac{2\pi}{b}\]where \( b \) is the frequency or the coefficient of \( \theta \).

• In \( y = \sin 5\theta \), the coefficient \( b \) is 5.
• Therefore, the period can be calculated using the formula: \( \frac{2\pi}{5} \).

This period means the function completes a cycle in as wide a range as \( \frac{2\pi}{5} \) or a little over 1.25 units on a horizontal graph. Shorter periods indicate that many cycles happen in a short span, which relates closely to oscillations and waves. Hence, this function repeats itself rapidly over smaller intervals.

Frequency

Frequency, in the context of trigonometric functions, refers to how many complete cycles the wave goes through over a certain interval. It's essentially the counterpart of the period.

• In \( y = \sin 5\theta \), the frequency is represented by the number 5, which is the coefficient of \( \theta \).
• This value indicates how many cycles occur in the full interval ranging from 0 to \( 2\pi \).

Thus, a frequency of 5 signifies exactly 5 complete cycles across this interval.Breaking down a sine wave into frequency can help understand how often the oscillation repeats, which is critical in various applications like signal processing or acoustics where sound waves and signals need to be analyzed for their repetitiveness and timing. By comprehending frequency, one can better appreciate the rhythmical nature of different phenomena modeled by trigonometric functions.