Question

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

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

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 \(R_2 = -4\).

Calculate amplitude

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

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.

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, \(y(t) = 5 \cos (2t + 0.93)\).

Key concepts

Amplitude Calculation

The amplitude in a trigonometric function indicates the peak value that the function reaches, representing its maximum displacement from the central axis. At the heart of amplitude calculation for a combination of sine and cosine functions is the Pythagorean Theorem. Here, to transform the function \(y(t) = 3 \cos 2t - 4 \sin 2t\), the amplitude \(A\) is found by determining the vector magnitude of the cosine and sine coefficients.

In this situation, the coefficients are 3 and -4 respectively. These act as the legs of a right triangle, where the hypotenuse is the amplitude \(A\). Thus, apply the formula:

• \(A = \sqrt{3^2 + (-4)^2}\)
• \(A = \sqrt{9 + 16}\)
• \(A = \sqrt{25} = 5\)

This shows that the function reaches up to 5 units above and below its central horizontal line.

Phase Shift

Phase shift refers to the horizontal translation of a wave function on a graph. It reveals how much the wave is shifted left or right from a standard position. In this exercise, discovering the phase shift \(\phi\) involves finding the inverse tangent or arctangent of the ratio of sine to cosine coefficients.

The formula used is:

• \(\phi = \arctan(-R_2/R_1)\)

Substituting the given values, we get:

• \(\phi = \arctan(-(-4)/3) = \arctan(4/3)\)
• The calculated value is approximately \(0.93\) radians.

This indicates that the function is shifted forward by about 0.93 units along the horizontal axis, altering its starting point along the graph.

Cosine and Sine Combination

Combining cosine and sine terms into a single trigonometric expression is a powerful technique in wave function analysis. The goal is to condense forms like \(3 \cos 2t - 4 \sin 2t\) into a unified expression. This unified version simplifies complex wave analysis and representation.

This transformation leverages the trigonometric identity that any sine and cosine combination can be expressed as a single cosine function with an appropriate amplitude and phase shift.

• \(y(t) = A \cos(2t + \phi)\)

Where \(A\), the amplitude, was calculated to be 5, and \(\phi\), the phase shift, calculated as approximately 0.93 radians. Therefore, the new expression becomes:

• \(y(t) = 5 \cos(2t + 0.93)\)

This transformation elegantly unifies the trigonometric identity into a single concise expression, providing simplicity and clarity in interpreting trigonometric functions.