Problem 5 - Essential Calculus

Question

Write the first five terms of the sequence. \(a_{n}=\sin \frac{n \pi}{2}\)

Step-by-Step Solution

Answer

The first five terms of the sequence are \(1, 0, -1, 0, 1\).

Step 1: Term 1

Substitute \(n=1\) into the function which gives \(a_{1}=\sin \frac{1 \pi}{2}\), and evaluating this expression gives \(a_{1}=1\)

Step 2: Term 2

Substitute \(n=2\) into the function to get \(a_{2}=\sin \frac{2 \pi}{2}\), this simplifies to \(a_{2}=\sin (\pi)\), evaluating this yields \(a_{2}=0\)

Step 3: Term 3

Substitute \(n=3\) into the function to get \(a_{3}=\sin \frac{3 \pi}{2}\), evaluating this yields \(a_{3}=-1\)

Step 4: Term 4

Substitute \(n=4\) into the function to get \(a_{4}=\sin \frac{4 \pi}{2}\), this simplifies to \(a_{4}=\sin (2\pi)\), evaluating this gives \(a_{4}=0\)

Step 5: Term 5

Substitute \(n=5\) into the function to obtain \(a_{5}=\sin \frac{5 \pi}{2}\), and upon evaluating this results in \(a_{5}=1\)

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in understanding periodic phenomena. They relate the angles of a triangle to the ratios of its sides. The sine function, denoted as \( \sin \theta \), is one of these functions.
It specifies the y-coordinate of a point on the unit circle corresponding to the angle \( \theta \).

Some key properties of the sine function include:

• Maximum and Minimum Values: The sine function oscillates between -1 and 1.
• Periodicity: The sine function is periodic with a period of \( 2\pi \). This means \( \sin(\theta + 2\pi) = \sin \theta \).
• Symmetry: It is an odd function, so \( \sin(-\theta) = -\sin \theta \).

In our exercise, the sine function determines the terms of the sequence \( a_{n} = \sin \frac{n\pi}{2} \). Understanding these properties helps us predict the values without full computation.

Sequences and Series

Sequences and series are mathematical concepts that involve ordered lists of numbers. A sequence is a set of numbers in which each number is derived according to a specific rule.
In our given exercise, the sequence is generated by a trigonometric rule: \( a_{n} = \sin \frac{n\pi}{2} \).lists:

• Terms: Each value in the sequence is called a term. In our exercise, terms are the values of \( \sin \frac{n\pi}{2} \) for \( n = 1, 2, 3, \dots \)
• Finite Sequences: A sequence with a fixed number of terms. Here, we computed the first five terms.
• Pattern Recognition: Sequences often have a recognizable pattern. By evaluating the first few terms in this sequence, you may notice a repeating cycle. Recognizing this cycle simplifies further predictions.

This cycle is crucial for efficiently handling exercises involving sequences.

Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting angles into trigonometric functions and simplifying. For the sequence \( a_{n} = \sin \frac{n\pi}{2} \), each term is evaluated by substituting successive integers into \( n \), giving specific angles.

Here's how evaluation typically works:

• Substitution: Replace the variable with a specific value. For instance, \( n = 1 \) gives \( \sin \frac{1\pi}{2} = 1 \).
• Simplification: Use known values of sine at key angles like \( \pi/2, \pi, 3\pi/2, \) and \( 2\pi \). This gives straightforward results without guesswork.
• Use of Symmetry and Periodicity: Applying properties of sine simplifies calculations. For example, knowing sine's periodicity helps to anticipate sequences repeating.

This understanding eases the process of finding and verifying the terms in a trigonometric sequence.