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Chapter 1: Problem 88

Identify the amplitude and period of the following functions. $$f(\theta)=2 \sin 2 \theta$$

Short Answer

Expert verified
Answer: The amplitude of the function is 2, and the period is $$\pi$$.

Step by step solution

01

Identify the amplitude

To find the amplitude of the given sinusoidal function, we need to look at the coefficient of the sin() function, which is the value in front of the sin(B胃). In our case, A = 2, so the amplitude is 2.
02

Identify the coefficient B

The coefficient B can be found inside the sin() function multiplying the angle 胃. In our case, the coefficient B is 2.
03

Find the period

Now we can find the period of the function using the formula $$T=\frac{2\pi}{B}$$ Substitute the value of B from Step 2 into the formula: $$T=\frac{2\pi}{2}$$ $$T=\pi$$ So, the amplitude of the given function is 2 and the period is $$\pi$$.

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Key Concepts

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

Amplitude
In trigonometric functions, understanding the amplitude is crucial as it represents the height of the wave from the centerline to the peak. For functions like the sine function, denoted as \( f(\theta) = A \sin(B\theta) \), the amplitude is directly identified as the value of \( A \). It indicates how far above and below the center line the wave reaches.
  • The amplitude is always a positive value.
  • It's determined by the coefficient in front of the sine or cosine term.
  • In our example, \( f(\theta) = 2 \sin 2\theta \), the amplitude is 2.
If you visualize a sine wave on a graph, the amplitude is the vertical stretch or compression. It doesn鈥檛 affect the length of the cycle but impacts the maximum and minimum values the function reaches.
Period
The period of a trigonometric function is the horizontal length it takes for the function to complete one full cycle before starting to repeat itself. For functions like \( f(\theta) = A \sin(B\theta) \), the period \( T \) can be calculated using the formula:
\[T = \frac{2\pi}{B}\]
  • The coefficient \( B \), found inside the function, dictates the speed of one cycle.
  • A larger \( B \) compresses the period, while a smaller \( B \) stretches it.
  • For the function \( f(\theta) = 2 \sin 2\theta \), substituting \( B = 2 \) gives a period of \( \pi \). This means every \( \pi \) units along the \( \theta \)-axis, the wave pattern repeats.
This concept helps to determine how frequently patterns repeat over a particular domain, essential for graphing and understanding wave behavior.
Sinusoidal Functions
Sinusoidal functions, such as sine and cosine, have unique properties making them ideal for modeling periodic phenomena like sound waves or tides. These functions take the form \( f(\theta) = A \sin(B\theta + C) + D \), where each component has a specific role:
  • \( A \) adjusts the amplitude, impacting the wave's height.
  • \( B \) alters the period, controlling how often cycles repeat.
  • \( C \) shifts the wave horizontally, known as the phase shift.
  • \( D \) moves the wave up or down vertically, adjusting the centerline.
These parameters allow sinusoidal functions to elegantly represent natural cycles. By tweaking these variables, the shape, position, and orientation of the wave can be tailored to match real-world data or desired outcomes. In the given example, the wave is described by \( f(\theta) = 2 \sin 2\theta \), indicating no phase shift or vertical displacement, making it a standard sine wave with a heightened amplitude.

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