Problem 68 - Precalculus with Calculus Previews

Question

Find the period of the given function. $$ f(x)=\sin \frac{3}{2} x+\cos \frac{5}{2} x $$

Step-by-Step Solution

Answer

The period of the function is \( 4\pi \).

Step 1: Identify the Frequencies of Each Trigonometric Function

The given function is a combination of a sine and a cosine function: \( f(x) = \sin\frac{3}{2}x + \cos\frac{5}{2}x \). For the sine function, \( \sin\frac{3}{2}x \), the coefficient of \( x \) is \( \frac{3}{2} \). For the cosine function, \( \cos\frac{5}{2}x \), the coefficient of \( x \) is \( \frac{5}{2} \). These coefficients determine the periods of the respective functions.

Step 2: Calculate the Period of Each Component

The standard period of \( \sin x \) and \( \cos x \) is \( 2\pi \). For \( \sin\frac{3}{2}x \), the period is calculated as follows: \( \text{Period of } \sin\frac{3}{2}x = \frac{2\pi}{\frac{3}{2}} = \frac{4\pi}{3} \). Similarly, for \( \cos\frac{5}{2}x \): \( \text{Period of } \cos\frac{5}{2}x = \frac{2\pi}{\frac{5}{2}} = \frac{4\pi}{5} \).

Step 3: Determine the Least Common Multiple (LCM) of the Periods

To find the period of the combined function, we need the least common multiple (LCM) of the individual periods: \( \frac{4\pi}{3} \) for \( \sin\frac{3}{2}x \) and \( \frac{4\pi}{5} \) for \( \cos\frac{5}{2}x \). Find the LCM using the denominators (3 and 5) and numerators (\(4\pi\), already common): - LCM of 3 and 5 is 15. - Thus, the LCM of the periods is \( \frac{4\pi \times 5}{15} = \frac{20\pi}{15} = \frac{4\pi}{3} \). - Similarly, the LCM with the other period aligns with \( \frac{20\pi}{3} \) and thus adjusts to \( 4\pi \).

Step 4: Confirm the Overall Period for the Function

Since \( 4\pi \) is a common multiple greater than the LCM calculated directly — both components repeating after an integer number of their periods (3 cycles of \( \frac{4\pi}{3} \) within \( 4\pi \) and 5 cycles of \( \frac{4\pi}{5} \) within \( 4\pi \)) — the actual period is \( 4\pi \).

Key Concepts

Periodicity

Periodicity is a fundamental property of trigonometric functions, allowing them to repeat their values at regular intervals. This concept makes sine and cosine functions incredibly useful for modeling repetitive phenomena, such as sound waves and pendulum motion. For any function, the period is the smallest positive number, say \( P \), for which the function repeats:

• Mathematically, this is expressed as \( f(x + P) = f(x) \) for all values of \( x \).

Understanding periodicity helps us determine how often a function's behavior cycles over time. In practice, different trigonometric functions can have different periods, and combinations of these can create complex, repeating output patterns. By identifying the periods of individual functions, we can infer the overall periodicity of a combined function.

Sine Function

The sine function, denoted as \( \sin(x) \), is a key trigonometric function with a standard period of \( 2\pi \). This means its values cycle every \( 2\pi \) units, repeating the wave-like pattern of its graph.

• The sine wave starts at zero, rises to a maximum of 1 at \( \pi/2 \), returns to zero at \( \pi \), hits a minimum of -1 at \( 3\pi/2 \), and returns to zero at \( 2\pi \).

The function's symmetry and periodicity enable it to model oscillatory behavior. When the sine function is altered by a coefficient as seen in \( \sin(\frac{3}{2}x) \), the period changes to \( \frac{4\pi}{3} \). Calculated as \( \frac{2\pi}{\text{frequency}} \), here the frequency is \( \frac{3}{2} \). Adjusting a function in this way changes how quickly it completes one full wave, allowing for control over the function's speed and frequency.

Cosine Function

Much like the sine function, the cosine function, denoted as \( \cos(x) \), also exhibits periodicity with a standard period of \( 2\pi \). Its graph shows a wave beginning at 1, descending to zero at \( \pi/2 \), reaching -1 at \( \pi \), back at zero at \( 3\pi/2 \), and concluding the cycle at 1 at \( 2\pi \).

• With changes in frequency, such as in \( \cos(\frac{5}{2}x) \), the period becomes \( \frac{4\pi}{5} \).

Similar to the sine function, its period is calculated as \( \frac{2\pi}{\text{frequency}} \) where the frequency here is \( \frac{5}{2} \). Adjusting the cosine function’s frequency alters how many cycles occur over a given interval. Together with sine, cosine modeling helps in simulating periodic processes, and in combination, they can synthesize various periodic patterns.

Least Common Multiple

The least common multiple (LCM) is an important concept used to find a common period in combined periodic functions, such as those involving sine and cosine. In the context of trigonometric functions

• Calculate the LCM of two periods to determine the minimal interval over which both functions repeat.

In our example function \( f(x) = \sin\frac{3}{2}x + \cos\frac{5}{2}x \), we first determined the individual periods: \( \frac{4\pi}{3} \) and \( \frac{4\pi}{5} \). The LCM involves:

• Finding the LCM of the denominators \( 3 \) and \( 5 \), which is \( 15 \).
• Multiplying the common numerators with the shared LCM denominator to find a combined period of \( 4\pi \).

Hence, the LCM provides a basis for calculating when two periodic functions will align again, offering a window into when all elements of a system reset or start their cycle anew.