5.4: Sinusoidal Models

Rewriting the Coordinate Definition

We begin this section with a brief interlude. Recall the coordinate definition of the trigonometric functions, which is presented below for our discussion.

If we hold \(r\) to be constant and plot all points a distance of \(r\) units from the origin, then we will have graphed a circle of radius \(r\) centered at the origin. By the coordinate definition of the trigonometric functions, if the point \(\left(x,y\right)\) is on this circle and is on the terminal side of \(\theta\), then\[\cos\left(\theta\right) = \dfrac{x}{r} \quad \text{and} \quad \sin\left(\theta\right) = \dfrac{y}{r}.\nonumber\]We have reached a point in this text where writing these formulas in an alternate form is more useful. Solving the formulas for \(x\) and \(y\), respectively, we get the following:\[x = r\cos\left(\theta\right) \quad \text{and} \quad y = r\sin\left(\theta\right).\label{polarform}\]The formulas in \ref{polarform} apply to any point on the terminal side of \(\theta\) on a circle of radius \(r\). Understanding these formulas is essential for our success moving forward in Trigonometry.

Sinusoidal Models

Now that we have a solid understanding of the graphs for the fundamental trigonometric functions, it's time we focused on their applications. In the same way exponential functions can be used to model a wide variety of phenomena in nature, the cosine and sine functions can be used to model their fair share of natural behaviors. In particular, sinusoidal waves are common in physics, engineering, computer science, signal processing, and many other fields. For consistency, we will focus on building applications using the sine function (although, you are welcome to use the Cofunction Identities to transform our sine models into cosine models). Moreover, we switch to the independent variable \(t\) (instead of \(x\) or \(\theta\)) since the applications in this section are time-dependent.

Before diving into the applications, let's summarize some important facts and definitions about the sine function. Let\[f(t) = A\,\sin\left(Bt + C\right) + D.\label{sinefnc}\]From the material so far in this chapter, we know that \(|A|\) is the amplitude, the period is \(\frac{2\pi}{B}\), the phase shift is \(-\frac{C}{B}\), and the vertical shift (or midline) is \(D\).

Along with knowing these formulas, it is helpful to remember what these quantities mean in context. As we discuss these contextual meanings, we will inevitably introduce more definitions (mostly from physics) that we will occasionally be using throughout the remainder of this text.

• The amplitude measures the sine wave's maximum displacement (change in position) from its midline. Therefore, if the midline of a sinusoidal function is \(y = 120\) and the amplitude is \(30\), the maximum displacement of the sinusoidal wave is \(30\).
• The period is the length of time it takes to complete one cycle of the sinusoid. For a slightly oversimplified example, the signal your antenna is receiving from a broadcasting radio station is a sinusoidal wave. If this wave takes \(1.0152 \times 10^{-8}\) seconds to complete a single cycle, then its period is \(1.0152 \times 10^{-8}\).
  • A more physically interesting, and important, interpretation related to the period is the ordinary frequency of the sinusoid (often just called the frequency). This was introduced previously but not really used, so let's recall some information about the frequency. This is the number of cycles the sinusoid goes through in a given unit of time (generally, per second). Since the period is in seconds per cycle and frequency is cycles per second, it should make sense that period and frequency are reciprocals. That is, if \(f\) is the frequency of a sinusoid and \(T\) is its period,\[f = \dfrac{1}{T} \quad \text{and} \quad T = \dfrac{1}{f}.\nonumber\]Thus, the frequency of the radio station mentioned above is\[f = \dfrac{1}{T} = \dfrac{1}{1.0152 \times 10^{-8}} \approx 98{,}502{,}758\,\text{cycles per second}.\nonumber\]The unit for "cycles per second" is called the Hertz (Hz). Therefore, the radio station is operating at roughly \(98.5\) million Hertz, or \(98.5\) MHz (megahertz).
  • Another concept we introduced previously is the angular frequency. In terms of Equation \ref{sinefnc}, the angular frequency is defined as the value of \(B\). This tells how many cycles are completed over an interval of length \(2\pi\).
• The phase shift represents how much of a "head start" the sinusoid has over the unshifted sine function.
  • The phase angle (also called the phase), which was only mentioned in a footnote in the previous section, is the value of \(C\). It is the angle corresponding to \(t = 0\). Since we have not officially written the definition for this, it is provided below.

All of this information is summarized below.

From this point forward, when dealing with applications, we will use the terms angular frequency, frequency, and phase when possible. Therefore, it's best to make a note of these definitions so you don't forget them.

Our first foray into sinusoidal motion puts these definitions and the equations in \ref{polarform} to good use.

A few remarks about Example \(\PageIndex{1}\) are in order. First, note that the amplitude of 64 in our answer corresponds to the radius of the Giant Wheel. This means that passengers on the Giant Wheel never stray more than 64 feet vertically from the center of the Wheel, which makes sense. Another way to say this (using our new definitions) is that the maximum displacement of a rider on the Great Wheel is 64 feet.

The second thing to note is our process of finding the value of the phase angle. Trigonometry is a fun topic because there are usually several ways to do something, and not one is better than another. In fact, if I were investigating this problem and were not concerned about instructing you on new language, I would have found the value of \(C\) using a different approach. To figure out the phase angle, we could have sketched the path of a rider on the Great Wheel from the time they enter \(\left(t = 0\right)\).

Notice that this sketch looks more like a reflected cosine than a sine; however, we made our choice to work with sine, and, darn it, we are going to stick with it. The actual sine function starts one-fourth of the way through this cycle. That is, at time \(\frac{1}{4}\left(\frac{127}{2}\right) = \frac{127}{8}\). Hence, the sinusoidal model has a phase shift of \(\frac{127}{8}\). Using the relation \(\text{Phase Shift} = -\frac{C}{B}\), we get\[\begin{array}{crcl} & \text{Phase Shift} & = & -\dfrac{C}{B} \\[6pt] \implies & \dfrac{127}{8} & = & -\dfrac{C}{4\pi/127} \\[6pt] \implies & \dfrac{127}{8} & = & -\dfrac{127C}{4\pi} \\[6pt] \implies & -\dfrac{4\pi}{8} & = & C \\[6pt] \implies & -\dfrac{\pi}{2} & = & C \\[6pt] \end{array}\nonumber\]Thus, our sinusoidal model is\[h(t) = 64\sin\left(\dfrac{4\pi}{127}t - \dfrac{\pi}{2}\right) + 72.\nonumber\]Now, you might say, "Hey, that's not what we got in Example \(\PageIndex{1}\)!" I beg to differ. Remember, the value of \(\sin\left(\theta\right)\) is the same as the value of \(\sin\left(\theta - 2\pi\right)\). If you subtract \(2\pi\) from the argument in the answer to Example \(\PageIndex{1}\), you would get \(h(t) = 64\sin\left(\dfrac{4\pi}{127}t - \dfrac{\pi}{2}\right) + 72\).

The third and final thing to mention is the phase shift of our answer, which is \(\frac{127}{8}=15.875\). This represents the "time delay" (in seconds) we introduce by starting the motion at the point \(P\) as opposed to the point \(Q\). Said differently, passengers who "start" at \(P\) take 15.875 seconds to "catch up" to the point \(Q\).

Our next example shows how to deal with data that is sinusoidal in nature.