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Chapter 5: Problem 6

Write \(y(t)=3 \cos 2 t-4 \sin 2 t\) in the form \(y(t)=A \cos (2 \pi f t+\phi)\)

Short Answer

Expert verified
The equation \(y(t)=3 \cos 2 t-4 \sin 2 t\) can be written in the form \(y(t)=5 \cos (2t + 0.93)\)

Step by step solution

01

Identify coefficients

In the given equation, the coefficients in front of cosine and sine are 3 and -4 respectively. Let us denote these as \(R_1\) and \(R_2\), giving \(R_1 = 3\) and \(R_2 = -4\).
02

Calculate amplitude

We calculate the ampliture \(A\) as the square root of the sum of squares of \(R_1\) and \(R_2\). So \(A = \sqrt{R_1^2 + R_2^2} = \sqrt{3^2 + (-4)^2} = 5\).
03

Calculate phase shift

The phase shift \(\phi\) is calculated by arc tangent of \(-R_2/R_1\). So \(\phi = \arctan (-R_2/R_1) = \arctan (-(-4)/3) = \arctan (4/3)\) which is approximately 0.93.
04

Formulate final equation

Now, we can write the original sine and cosine combination as a single cosine function with amplitude, frequency and phase shift. Therefore, \(y(t) = 5 \cos (2t + 0.93)\).

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