Problem 37 - 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. $$ y=3 \sin \theta $$

Step-by-Step Solution

Answer

The function \(y = 3\sin(\theta)\) has one cycle in the interval from 0 to \(2\pi\). The amplitude is 3 and the period is \(2\pi\).

Step 1: Identify the number of cycles

A complete cycle in sine function within the interval from 0 to \(2\pi\) is called a period. For the function \(y = 3\sin(\theta)\), within the interval from 0 to \(2\pi\) it completes one full cycle.

Step 2: Identify the amplitude

The amplitude of a sine function is the absolute value of the coefficient of the sin function. In this function \(y = 3\sin(\theta)\), the coefficient of \(\sin(\theta)\) is 3. Therefore, the amplitude is the absolute value of 3, which is 3.

Step 3: Identify the period

The period of a sine function is the length of one complete cycle. For a normal sine function, the period is equal to \(2\pi\). For the function \(y = 3\sin(\theta)\), its parent function is the normal sine function. Therefore, the period is also \(2\pi\).

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions and is known for its wave-like pattern. It is often used to model periodic phenomena like sound waves or tides. The basic form of the sine function is given as:

• \( y = \sin(\theta) \)

Here, \( \theta \) represents the angle, usually in radians, and \( y \) is the sine value of \( \theta \). The sine function has a smooth, repetitive pattern that oscillates above and below the horizontal axis, which is why it's called a periodic function.
The graph of the sine function begins at the origin (0,0), rises to a maximum, descends through the origin again, reaches a minimum, and then returns to the origin to complete one full cycle. This characteristic shape is known as the sine wave. Its smooth and continuous wave pattern makes it distinct among trigonometric functions.

Amplitude

The amplitude of a sine function is a measure of the height of its peaks and the depths of its troughs from its central axis. Specifically, it represents the maximum absolute value of the function from its median line. Mathematically, the amplitude is determined by the coefficient in front of the sine function in its equation.For the function, \( y = 3 \sin(\theta) \), the coefficient is 3. The amplitude is the absolute value of this coefficient, which is 3. This means the graph of this function will rise 3 units above and fall 3 units below its central axis. The larger the amplitude, the taller and deeper the oscillations of the sine wave will be, giving it more "intensity" in its pattern.Understanding amplitude is crucial when analyzing wave-like patterns, as it tells you the range of the oscillation.

Period

The period of a sine function is the distance required along the horizontal axis for the function to complete one full cycle. In simpler terms, it is the length of the smallest interval after which the wave pattern repeats itself. For the standard sine function \( y = \sin(\theta) \), the period is \( 2\pi \), which means it takes \( 2\pi \) radians for the sine wave to complete one full cycle.For modifications of the sine function, such as \( y = a\sin(b\theta) \), the period changes according to the formula:

• Period = \( \frac{2\pi}{b} \)

In the function \( y = 3\sin(\theta) \), \( b \) equals 1, thus maintaining the standard period of \( 2\pi \). The consistency of this repetition helps in predicting and modeling periodic behavior in practical applications.

Cycle

A cycle in the context of the sine function refers to a complete set of movements from a starting value, through a maximum, to a minimum, and back to the starting value. This is sometimes also referred to as a "wave cycle" or "oscillation." For the function \( y = 3\sin(\theta) \), a single cycle is completed as the angle \( \theta \) goes from 0 to \( 2\pi \).For any sine function, one full cycle involves moving from the centerline to a peak, back through the centerline to a trough, and back to the centerline. This cycle pattern mirrors what's actually occurring with many natural processes, from sound vibrations to ocean waves.
Understanding cycles is fundamental when analyzing wave patterns, as it helps predict when and how often these patterns repeat over a given time or space.