Problem 35 - Calculus for Biology and Medicine

Question

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ \sin (\pi), \sin (2 \pi), \sin (3 \pi), \sin (4 \pi), \sin (5 \pi), \ldots $$

Step-by-Step Solution

Answer

The expression for \( a_n \) is \( a_n = 0 \) for all integers \( n \).

Step 1: Understand the Sine Function

The sine function, \( \sin(x) \), is periodic with a period of \( 2\pi \). This means that \( \sin(x + 2\pi k) = \sin(x) \) for any integer \( k \). In this problem, we need to evaluate the sine function at integer multiples of \( \pi \).

Step 2: Evaluate the Sine at Integer Multiples of \( \pi \)

Compute the sine values at different multiples of \( \pi \): \[ \begin{aligned} \sin(\pi) &= 0, \ \sin(2\pi) &= 0, \ \sin(3\pi) &= 0, \ \sin(4\pi) &= 0, \ \sin(5\pi) &= 0, \ldots \end{aligned} \] The pattern is there; for any integer \( n \), \( \sin(n\pi) = 0 \).

Step 3: Write the Expression for \( a_n \)

Since \( \sin(n\pi) = 0 \) for any integer \( n \), the expression for \( a_n \) will be a constant zero: \[ a_n = 0 \] for all integers \( n \).

Step 4: Conclusion

Every term in the sequence of \( a_n \) values is evaluated as zero. Therefore, regardless of which term or index, the function can be replaced by zero since \( a_n = \sin(n\pi) = 0 \) across the sequence.

Key Concepts

Trigonometric Functions

Trigonometric functions are essential in mathematics, especially in the study of angles and periodic phenomena. These functions, such as sine, cosine, and tangent, describe the relationships between the angles and sides of a right triangle. As periodic functions, they repeat their values at regular intervals. This property is helpful in analyzing wave patterns, sound vibrations, and even the orbits of celestial objects.

One of the most crucial aspects of trigonometric functions is their periodicity:

• Sine and Cosine: Both have a period of \(2\pi\), meaning they repeat every \(2\pi\) units.
• Tangent: This function has a period of \(\pi\), so it repeats its pattern every \(\pi\) units.

Understanding these periods helps in predicting values of the functions over different domains.

Sine Function

The sine function, denoted as \( \sin(x) \), is a fundamental trigonometric function. It maps every real number to a value between -1 and 1. When considering the unit circle, the sine of an angle represents the y-coordinate of the point corresponding to that angle.

• In the context of the unit circle, \( \sin(0) = 0 \), \( \sin(\pi) = 0 \), and \( \sin(2\pi) = 0 \), etc.
• This property reflects its zero points at integer multiples of \(\pi\).

A core property of the sine function is its periodicity, repeating every \(2\pi\). This means that for any integer \(k\), the equation \( \sin(x + 2\pi k) = \sin(x) \) holds true.

This understanding is crucial in many areas, such as physics and engineering, where waveforms and oscillations are modeled using the sine function.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a specific pattern. Understanding sequences can help in grasping more complex mathematical concepts, such as series and integrals. An important aspect of sequences is finding a general formula, which describes each term's behavior within the sequence.

For the sine values given in the exercise, the sequence \( a_n = \sin(n\pi) \) simplifies to:

• Initial Terms: Consider the terms \( \sin(\pi), \sin(2\pi), \sin(3\pi) \), which all equal zero.
• Pattern Recognition: Observing that the sine function is zero at every integer multiple of \(\pi\).

The expression for \(a_n\) can be written as a constant value of zero, \( a_n = 0 \), because every term in this particular sequence simplifies to zero, demonstrating a simple pattern and uniformity across terms.