Question

determine \(\omega_{0}, R,\) and \(\delta\) so as to write the given expression in the form \(u=R \cos \left(\omega_{0} t-\delta\right)\) $$ u=3 \cos 2 t+4 \sin 2 t $$

Short answer

Question: Rewrite the given function \(u = 3 \cos 2t + 4 \sin 2t\) in the form of \(u = R \cos(\omega_0 t - \delta)\). Answer: \(u = 5 \cos(2t - \arctan(\frac{4}{3}))\)

Step-by-step solution

Write the given function using a single trigonometric function

Using the sum-to-product identity, we can rewrite the given function as a product of a single trigonometric function: $$ u = A \cos 2t + B \sin 2t $$ Notice that this is the same as the original function with \(A=3\) and \(B=4\).

Use the angle addition formula to find the desired form

We want to express the function in the form of \(u = R \cos(\omega_0 t - \delta)\). To do this, we will apply the angle addition formula for cosine: $$ \cos(\omega_0 t - \delta) = \cos(\omega_0 t) \cos(\delta) + \sin(\omega_0 t) \sin(\delta) $$ Now, we will identify the appropriate values for \(\omega_0\), \(R\), \(\delta\), \(A\), and \(B\), by comparing the two expressions. First, notice that both expressions have the same angular frequency (i.e., the coefficient of \(t\)), so we have \(\omega_0 = 2\).

Find the amplitude (R) and phase shift (\(\delta\))

We can rewrite the previous step as: $$ u = R[\cos(\omega_0 t)\cos(\delta) + \sin(\omega_0 t)\sin(\delta)] $$ Now we compare coefficients of \(\cos 2t\) and \(\sin 2t\) in both expressions: \(A = R\cos\delta\) and \(B = R\sin\delta\). Given \(A=3\) and \(B=4\), we can use the Pythagorean theorem to get the amplitude: $$ R = \sqrt{A^2 + B^2} = \sqrt{3^2 + 4^2} = 5 $$ To find the phase shift, divide the second equation by the first equation: $$ \tan\delta = \frac{B}{A} = \frac{4}{3} $$ Taking the arctangent of both sides, we get \(\delta = \arctan(\frac{4}{3})\).

Write the final expression

Now that we have found the values of \(\omega_0\), \(R\), and \(\delta\), we can write the final expression: $$ u = R\cos(\omega_0 t - \delta) = 5\cos(2t - \arctan(\frac{4}{3})) $$ So the given expression can be written in the desired form as \(u = 5 \cos(2t - \arctan(\frac{4}{3}))\).

Key concepts

Sum-to-Product Identity

The sum-to-product identities are a set of trigonometric identities that convert the sum or difference of two trigonometric functions into a product of trigonometric functions. This identity is particularly useful when simplifying trigonometric expressions or solving trigonometric equations. To understand this concept, let's delve deeper into one of the sum-to-product identities:

For sine and cosine functions, the identity is given by:
\[ \cos A + \cos B = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right) \]
and
\[ \sin A + \sin B = 2\sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right) \]

In the textbook exercise, the sum-to-product identity wasn't used directly to combine the sine and cosine terms into a single product because the goal was to express the function using a single trigonometric function, specifically a cosine function with a phase shift. However, understanding this identity can aid in recognizing patterns and strategies for manipulating trigonometric expressions.

Angle Addition Formula

The angle addition formulas allow us to expand trigonometric functions of summed angles into products of trigonometric functions of single angles. These formulas are essential in many areas of mathematics including calculus and can be seen as:
\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]
and
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]

In the provided example, the angle addition formula is used in reverse to find a cosine function with a phase shift that represents the sum of a sine and a cosine function. To clarify, if we have an expression like \( A \cos \theta + B \sin \theta \), we can rewrite it as a single cosine function by identifying it with the expansion of the cosine angle addition formula, effectively finding an equivalent expression that describes a single wave motion with a phase shift.

Trigonometric Phase Shift

Trigonometric phase shift describes the horizontal shift of trigonometric functions on a graph. It dictates how much a function is shifted to the left or right from its standard position. In the context of our example, a function of the form \( R \cos(\omega_0 t - \delta) \) represents a cosine wave with amplitude \( R \), angular frequency \( \omega_0 \), and a phase shift of \( \delta \). Here, the phase shift \( \delta \) moves the entire wave to the right or left along the horizontal axis by \( \delta \) units. The process of finding the phase shift involves comparing and then solving equations to relate the shift to the original function components, as we saw in the calculation involving the arctangent function to determine the value of \( \delta \).

Angular Frequency

Angular frequency is a measure of how rapidly an object rotates or how frequently a wave oscillates in terms of radians per unit time. Specifically, it is the rate at which the angle (in radians) changes over time and is denoted as \( \omega \) (omega). For trigonometric functions that describe waves, such as \( u = R \cos(\omega_0 t - \delta) \), angular frequency is the coefficient of \( t \) in the argument of the cosine function. This value indicates the number of radians the wave cycles through per unit of time. In the example given, the angular frequency is \( \omega_0 = 2 \), meaning the wave completes 2 radians (or about 114.6 degrees) of its cycle each unit of time. It's a key parameter in understanding wave behavior and is directly related to the period and frequency of the wave.