Question

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

Short answer

Q: Rewrite the function \(u=4 \cos 3t - 2 \sin 3t\) in the form \(u=R \cos(\omega_{0} t - \delta)\) where R, \(\omega_0\), and \(\delta\) are parameters. A: The given function can be rewritten in two possible ways: 1) \(u = 4\cos(3t - m\pi)\), where m is any integer 2) \(u = 2\cos\left(3t - \left(2m + 1\right)\frac{\pi}{2}\right)\), where m is any integer

Step-by-step solution

Identify the given function and target function

We are given the function: $$ u=4 \cos 3t - 2 \sin 3t $$ We want to rewrite it in the form: $$ u=R \cos(\omega_{0} t - \delta) $$ where we need to find the values of \(\omega_0, R,\) and \(\delta\).

Use the trigonometric identity for sum of angles

Recall the trigonometric identity for the cosine of a sum of angles: $$ \cos (A - B) = \cos A \cos B + \sin A \sin B $$ We can compare this identity to our given function, and identify that: $$ A = \omega_{0}t, \quad B = \delta, \quad \cos A \cos B = 4 \cos 3t, \quad \sin A \sin B = -2 \sin 3t $$

Set up a system of equations

We can write a system of equations based on the trigonometric identity and our given function using the relations identified in Step 2: \begin{align*} R \cos \omega_{0} t \cos \delta &= 4 \cos 3t\\ R \sin \omega_{0} t \sin \delta &= -2 \sin 3t \end{align*} As we see that both the equations are identical except that sine function is replaced by cosine function and vice versa, we are only left with finding the values of \(R\) and \(\delta\).

Calculate R using Pythagorean identity

The Pythagorean identity states that: $$ \cos^{2} A + \sin^{2} A = 1 $$ Using the relations identified in Step 2, we can rewrite this identity as: $$ \left( \frac{4}{R \cos \delta} \right)^2 + \left( \frac{-2}{R \sin \delta} \right)^2 = 1 $$ Simplifying and solving for R: \begin{align*} 16 \frac{\cos^2 \delta}{R^2} + 4 \frac{\sin^2 \delta}{R^2} &= 1\\ 16 \cos^2 \delta + 4 \sin^2 \delta &= R^2 \\ R &= \sqrt{16\cos^2 \delta + 4 \sin^2 \delta} \end{align*}

Calculate \(\delta\) using the tangent function

Divide the second equation from the system of equations (Step 3) by the first one: \begin{align*} \frac{R \sin \omega_{0} t \sin \delta}{R \cos \omega_{0} t \cos \delta} &= \frac{-2 \sin 3t}{4 \cos 3t}\\ \tan \left(\omega_{0} t + \delta\right) &= \frac{-2 \sin 3t}{4 \cos 3t} \end{align*} Since \(\omega_0 t = 3t\), we can rewrite the expression as: $$ \tan(3t + \delta) = -\frac{1}{2} \tan 3t $$ As the tangent function is odd, we can simplify this equation as follows: $$ \tan(3t + \delta) = \tan \left(3t - \delta\right) $$ This means the angles \(3t + \delta\) and \(3t - \delta\) have the same tangent, which is only possible when their sum is a multiple of \(\pi\), i.e., when $$ \delta = \frac{n \pi}{2} $$ where n is an integer.

Plug \(\delta\) back into the expression for R

Plugging the expression for \(\delta\) back into the expression for \(R\), we get: $$ R = \sqrt{16\cos^2 \left(\frac{n \pi}{2}\right) + 4 \sin^2 \left(\frac{n \pi}{2}\right)} $$ If \(n\) is even (say, \(n=2m\)), then we get: \begin{align*} R &= \sqrt{16\cos^2 (m \pi) + 4 \sin^2 (m \pi)}\\ &= \sqrt{(16)(1) + (4)(0)}\\ &= 4 \end{align*} And if \(n\) is odd (say, \(n=2m+1\)), then we get: \begin{align*} R &= \sqrt{16\cos^2 \left(\frac{(2m+1) \pi}{2}\right) + 4 \sin^2 \left(\frac{(2m+1) \pi}{2}\right)}\\ &= \sqrt{(16)(0) + (4)(1)}\\ &= 2 \end{align*} So, we have two possibilities for \(R\) and \(\delta\): 1) \(R=4\), and \(\delta = m\pi\) (for any integer m) 2) \(R=2\), and \(\delta = (2m+1)\frac{\pi}{2}\) (for any integer m) Therefore, the expression can be written as: $$ u = 4\cos(3t - m\pi) \quad \text{or} \quad u = 2\cos\left(3t - \left(2m + 1\right)\frac{\pi}{2}\right) $$ Where m is any integer.

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are powerful tools for modeling a wide range of phenomena in science and engineering, such as motion, heat, and growth. A simple example of a differential equation is Newton's second law of motion, which can be written as \( F = ma \) where \( F \) is the force applied to an object, \( m \) is the mass, and \( a \) is the acceleration (or the second derivative of position with respect to time).

When working with trigonometric functions and differential equations, it's common to transform complex expressions into a more manageable form. This can involve converting between different trigonometric representations or applying identities to simplify an expression, making it easier to integrate or differentiate as required in solving differential equations.

Boundary Value Problems

Boundary value problems (BVPs) are specific types of differential equations where you're given not only the equation to solve but also the values (the 'boundary' conditions) that the solution needs to satisfy at the extremes of the domain. For instance, in a physical problem, these might represent the condition of a system at the start and end of a time period, or at two different positions in space.

These problems are common in the real world, for example, in calculating the temperature distribution in a rod at steady state, predicting the steady-state vibration of a beam, or when determining the electric potential in a region given potentials on the boundary. To solve BVPs, multiple techniques are used including analytical methods like separation of variables, and numerical methods such as finite differences and shooting methods.

Trigonometric Identity

Trigonometric identities are equations that relate the trigonometric functions (sine, cosine, tangent, etc.) to one another. They are foundational in solving many problems involving triangles and oscillating functions. An example of a fundamental trigonometric identity is the Pythagorean identity, \( \cos^2 A + \sin^2 A = 1 \), which states that for any angle \( A \), the square of the cosine plus the square of the sine of that angle equals one.

Trigonometric identities are especially useful in analysis and calculus for transforming a trigonometric expression into an equivalent form that is more convenient for a particular purpose. In the context of differential equations and boundary value problems, these identities can be used to manipulate and solve equations that involve trigonometric functions.

System of Equations

A system of equations is a set of multiple equations that you solve together because they have shared variables. Solving a system of equations requires you to find the values of the variables that make all the equations hold true simultaneously. The methods used for solving systems of equations include substitution, elimination, and graphical representation.

With regard to trigonometric functions, systems of equations often arise when we deal with conversions from one form to another, as we see with converting a trigonometric expression into its trigonometric form conversion. In our example, by employing trigonometric identities and comparing coefficients, a system is established to determine the specific values for the amplitude, frequency, and phase shift.