Problem 7 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

Find the periods of the following functions. \(\begin{array}{ll}\text { a. } & P(t)=\sin \left(\frac{\pi}{3} t\right) \\\ \text { b. } & P(t)=\sin (t) \\ \text { c. } & P(t)=5-2 \sin (t)\end{array}\) \(\begin{array}{ll}\text { d. } & P(t)=\sin (t)+\cos (t) \\ \text { e. } & P(t)=\sin \left(\frac{2 \pi}{2} t\right)+\sin \left(\frac{2 \pi}{3} t\right) \\\ \text { f. } & P(t)=\tan 2 t\end{array}\)

Step-by-Step Solution

Answer

a. 6, b. 2π, c. 2π, d. 2π, e. 6, f. π/2.

Step 1: Period of a Sine Function

For the function \( P(t) = \sin\left(\frac{\pi}{3} t\right) \), the period can be found using the formula \( T = \frac{2\pi}{b} \) where \( b = \frac{\pi}{3} \). Therefore, \( T = \frac{2\pi}{\frac{\pi}{3}} = 6 \).

Step 2: Standard Sine Function Period

For \( P(t) = \sin(t) \), the period of the standard sine function is \( 2\pi \).

Step 3: Period of a Sine Function with a Constant

For \( P(t) = 5 - 2\sin(t) \), the constant and coefficient only affect the amplitude and vertical shift, not the period. Thus, the period remains \( 2\pi \).

Step 4: Period of Sum of Sine and Cosine Function

For \( P(t) = \sin(t) + \cos(t) \), the period is determined by the greatest common period of both functions. Since both \( \sin(t) \) and \( \cos(t) \) have a period of \( 2\pi \), the period of the sum is also \( 2\pi \).

Step 5: Period of Sum of Two Sine Functions with Different Frequencies

For \( P(t) = \sin\left(\frac{2\pi}{2} t\right) + \sin\left(\frac{2\pi}{3} t\right) \), calculate periods separately: \( T_1 = \frac{2\pi}{\frac{2\pi}{2}} = 2 \) and \( T_2 = \frac{2\pi}{\frac{2\pi}{3}} = 3 \). The common period is the least common multiple, LCM(2, 3) = 6.

Step 6: Period of a Tangent Function

For \( P(t) = \tan(2t) \), the period of a tangent function is \( T = \frac{\pi}{b} \), where \( b = 2 \). Thus, \( T = \frac{\pi}{2} \).

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry, representing a smooth periodic wave. It measures the vertical component of an angle's rotation on the unit circle. The standard formula for the sine function is expressed as \( \sin(t) \), which naturally exhibits a period of \( 2\pi \). This means that the function repeats its pattern every \( 2\pi \) units of time or angle. The period can change when the angle within the sine function is adjusted. For example, if you have \( \sin\left(\frac{\pi}{3}t\right) \), the period \( T \) becomes \( \frac{2\pi}{\frac{\pi}{3}} = 6 \). This scaling factor compresses or stretches the wave along the horizontal axis. Hence, the period is essential in understanding the behavior of the sine function over its cycle.

Cosine Function

Much like the sine function, the cosine function is another pivotal trigonometric function. It measures the horizontal component of an angle's rotation on the unit circle and is given by \( \cos(t) \). The cosine function, interestingly, also has a standard period of \( 2\pi \). Similar to its sine counterpart, adjustments within the cosine function, such as changes to the angle or frequency, can alter its period. For combinations of sine and cosine functions, like in a sum \( \sin(t) + \cos(t) \), it's critical to determine the greatest common period. Since both \( \sin(t) \) and \( \cos(t) \) naturally complete a cycle every \( 2\pi \), their sum will share this period, maintaining the same pattern repetition over \( 2\pi \).

Tangent Function

The tangent function, represented as \( \tan(t) \), differs from sine and cosine by having a recurring cycle of \( \pi \) due to its vertical asymptotes and undefined points where cosine is zero. This results in a pattern that repeats every \( \pi \) units, making it unique among the basic trigonometric functions. When the tangent function is adjusted, such as in \( \tan(2t) \), the period becomes \( \frac{\pi}{2} \). This indicates the function repeats its behavior over half of its standard cycle length. Understanding the period is helpful, especially when modeling waves or vibrations using the tangent function.

Least Common Multiple

The least common multiple (LCM) is a concept that comes into play when determining a common period for functions of different frequencies. For example, if you have two sine functions like \( \sin\left(\frac{2\pi}{2}t\right) \) and \( \sin\left(\frac{2\pi}{3}t\right) \) with periods 2 and 3 respectively, the LCM helps find the shared repeating interval. The LCM of 2 and 3 is 6, indicating that the combined function will have a cycle that repeats every 6 units. This mathematical tool is crucial in synthesizing periodic functions, ensuring predictability and synchronization in various applications such as signal processing and harmonics in physics.