Question

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

Short answer

Question: Rewrite the expression \(u = -2 \cos \pi t - 3 \sin \pi t\) in the form \(u = R \cos(\omega_{0} t - \delta)\), and find the values of \(R\), \(\omega_{0}\), and \(\delta\). Answer: The rewritten expression is \(u = \sqrt{13}\cos\left(\pi t-\arcsin\left(\frac{-3}{\sqrt{13}}\right)\right)\), where \(R = \sqrt{13}\), \(\omega_{0} = \pi\), and \(\delta=\arcsin\left(\frac{-3}{\sqrt{13}}\right)\).

Step-by-step solution

Find the frequency \(\omega_{0}\)

To rewrite the given expression in the desired form, we need to find the frequency \(\omega_{0}\). Looking at the given expression, we can see that both trigonometric terms have the same frequency, \(\pi\). Therefore, the frequency we are looking for is \(\omega_{0} = \pi\).

Rewrite the expression using the trigonometric identity

We will use the trigonometric identity \(\cos(x - y) = \cos(x)\cos(y) + \sin(x)\sin(y)\) to rewrite the given expression. To do this, we simply need to match the coefficients of the cosine and sine terms and find the angle \(\delta\). Our expression becomes: $$ u=R\cos(\pi t-\delta) = R(\cos(\pi t)\cos(\delta) + \sin(\pi t)\sin(\delta)) $$

Compare coefficients and find the amplitude R

Now, we need to match the coefficients of the same terms on both sides of the equation. After comparing the coefficients, we get the following equations: $$ -2 = R\cos(\delta) \\ -3 = R\sin(\delta) $$ We now find the amplitude R by squaring and adding these two equations: $$ (-2)^2 + (-3)^2 = R^2(\cos^2(\delta) + \sin^2(\delta)) $$ Simplifying the equation and using the trigonometric identity \(\cos^2(\delta) + \sin^2(\delta) = 1\), we get: $$ R^2= 4 + 9 = 13 $$ So, the amplitude is \(R = \sqrt{13}\).

Calculate the phase angle \(\delta\)

We can now use one of the equations we found earlier to calculate the phase angle \(\delta\). Let's use the equation \(-3 = R\sin(\delta)\). We plug in the amplitude \(R = \sqrt{13}\) and solve for \(\delta\): $$ -3 = \sqrt{13}\sin(\delta) $$ Dividing both sides by \(\sqrt{13}\), we get: $$ \sin(\delta) = \frac{-3}{\sqrt{13}} $$ Taking the inverse sine, we get: $$ \delta=\arcsin\left(\frac{-3}{\sqrt{13}}\right) $$

Rewrite the expression with the found values

We have found the values for \(\omega_{0}, R\), and \(\delta\). Now we can rewrite the given expression in the desired form: $$ u = \sqrt{13}\cos\left(\pi t-\arcsin\left(\frac{-3}{\sqrt{13}}\right)\right) $$ This is the final expression in the form \(u=R \cos \left(\omega_{0} t-\delta\right)\).

Key concepts

Amplitude and Phase Angle

In trigonometry, amplitude is a term used to describe the maximum extent of a vibration or oscillation, measured from the position of equilibrium. When we are dealing with waves or periodic functions, amplitude is essentially the 'height' of the wave, or in this case, the coefficient preceding the trigonometric function.
For example, in the exercise, we determine the amplitude of a cosine function. Mathematically, this is expressed as R in the general form of the wave equation, which is represented as:

\[ u = R\cos(\omega_0 t - \delta) \]

This equation is also common in physics to describe harmonic motion which we'll discuss later. The square root of the sum of the squares of the coefficients in front of the cosine and sine terms gives us the amplitude, representing the maximum displacement from rest.
When tackling the phase angle, \( \delta \) in our case, it's the horizontal shift of the wave from its origin. This value determines where the wave starts and changes how the function behaves over time. In the solution, we calculated the phase angle using the inverse sine function of the ratio obtained by dividing the coefficient in front of the sine term by the calculated amplitude.

Harmonic Motion

Harmonic motion, especially simple harmonic motion (SHM), is a fundamental concept in physics describing a type of periodic oscillation that is restorative and sinusoidal. In other words, it's the motion of an object that returns to a mean position in response to a restoring force, like a mass on a spring or a pendulum swinging.
In our trigonometric function, the notion of harmonic motion is encapsulated in the form of the cosine wave. In such motion, the amplitude signifies the maximum displacement, the angular frequency \( \omega_0 \) relates to how many oscillations occur per unit time, and the phase angle \( \delta \) indicates the initial position of the motion at \( t = 0 \) (the starting point in time).
A simple harmonic oscillator can be mathematically represented by a differential equation, which leads us to its connection with our next core concept. Understanding this relationship with differential equations is crucial for solving complex problems in fields ranging from mechanical engineering to quantum physics.

Differential Equations

In mathematics, differential equations are equations that contain functions and their derivatives. These equations are instrumental in describing various phenomena such as growth and decay, the flow of heat, or motion. In our context, the motion of a wave or oscillatory system can often be modeled by a differential equation.
In simple harmonic motion, for example, the restoring force, and thus the acceleration of an object, is proportional to the displacement of the object from its equilibrium position but in the opposite direction. The mathematical representation of this is a second-order linear differential equation.
To connect back to the textbook example, a function that describes SHM, like our \( u = R\cos(\omega_0 t - \delta) \) equation, can be derived from such a differential equation. The steps taken in the solution essentially involved reverse-engineering the constants within a trigonometric function, typically obtained by solving a differential equation governing the harmonic motion. The process demonstrates the practical application of these equations in analyzing and interpreting the physical phenomena they represent.