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Chapter 5: Problem 24

Is the function \(f(x)=5 \cos x+3 \sin x\) sinusoidal? If it is sinusoidal, state the period of the function.

Short Answer

Expert verified
Yes, the function is sinusoidal with period \(2\pi\).

Step by step solution

01

- Identify the given function

We are given the function: \[ f(x) = 5 \cos x + 3 \sin x \]We need to determine if this function is sinusoidal and find its period if it is.
02

- Define a sinusoidal function

A sinusoidal function is typically in the form of \[ A \cos(Bx + C) + D \] or \[ A \sin(Bx + C) + D \]where \(A\), \(B\), \(C\), and \(D\) are constants.
03

- Rewrite the given function using a single sinusoidal function

To determine if the function is sinusoidal, we can rewrite the given function \[ f(x) = 5 \cos x + 3 \sin x \] in the form of \[ R \cos(x - \theta) \]by using the trigonometric identities.
04

- Using the trigonometric identity

Using the trigonometric identity, we let \[ R \cos(x - \theta) = R \cos x \cos \theta + R \sin x \sin \theta \]for some \(R\) and \(\theta\). By comparing terms,\[ R \cos \theta = 5 \]\[ R \sin \theta = 3 \]
05

- Solve for \(R\)

To find \(R\), use the Pythagorean identity:\[ R = \sqrt{(R \cos \theta)^2 + (R \sin \theta)^2} = \sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34} \]
06

- Confirm the identity

We can rewrite the function as: \[ f(x) = \sqrt{34} \left( \frac{5}{\sqrt{34}} \cos x + \frac{3}{\sqrt{34}} \sin x \right) \]This validates the given function as a sinusoidal function.
07

- Find the period of the function

A standard sinusoidal function \( \cos(Bx + C) \) or \( \sin(Bx + C) \) has period \[ \frac{2\pi}{B} \]In this case, \( B = 1 \), so the period is\[ \frac{2\pi}{1} = 2\pi \]

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Key Concepts

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

Sinusoidal Functions
Sinusoidal functions are mathematical functions that produce a wave-like graph. They include sine and cosine functions. These functions are essential in modeling oscillations and waves.

A general sinusoidal function is either of the form \(A \cos(Bx + C) + D \) or \(A \sin(Bx + C) + D \), where:

  • \

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Most popular questions from this chapter

Determine the period, in radians, of each function using two different methods. a) \(y=-2 \sin 3 x\) b) \(y=-\frac{2}{3} \cos \frac{\pi}{6} x\)

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