Problem 4 - University Calculus: Early Transcendentals

Question

Without sigma notation. Then evaluate them. $$\sum_{k=1}^{5} \sin k \pi$$

Step-by-Step Solution

Answer

The sum is 0.

Step 1: Understanding the Sum

We need to evaluate the sum \(\sum_{k=1}^{5} \sin(k \pi)\), which means calculating the value of \(\sin(k \pi)\) for each integer \(k\) from 1 to 5, then summing these values together.

Step 2: Evaluating Individual Terms

The sine function at integer multiples of \(\pi\) takes specific values. Specifically, \(\sin(n \pi) = 0\) for any integer \(n\). Hence, \(\sin(1 \pi) = 0\), \(\sin(2 \pi) = 0\), \(\sin(3 \pi) = 0\), \(\sin(4 \pi) = 0\), and \(\sin(5 \pi) = 0\).

Step 3: Summing the Values

Now, we sum the values obtained: \(0 + 0 + 0 + 0 + 0 = 0\).

Step 4: Final Result

The result of the sum \(\sum_{k=1}^{5} \sin(k \pi)\) is \(0\).

Key Concepts

Trigonometric Functions

Trigonometric functions are an essential part of mathematics, often used to relate the angles of a triangle to the lengths of its sides. The most common trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). These functions are periodic and are based on the unit circle concept. For any angle, these functions help identify the corresponding point on the unit circle.

• The sine function, represented as \( \sin \theta \), is the y-coordinate of the point on the unit circle corresponding to the angle \( \theta \).
• It is periodic with a period of \( 2\pi \), which means that \( \sin(\theta + 2\pi) = \sin \theta \)
• Trigonometric functions have defined values for key angles such as \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \), and multiples thereof.

Understanding these functions' periodic nature and specific angle values is crucial for evaluating trigonometric expressions efficiently.

Integer Multiples of Pi

When dealing with trigonometric functions, special consideration is given to integer multiples of \( \pi \). In the context of sine and cosine, these multiples simplify evaluation drastically. The reason is that these points correspond to angles on the unit circle where the sine and cosine take on simple values.

• For any integer \( n \), \( \sin(n\pi) = 0 \). This is because these points lie on the x-axis of the unit circle, where the y-coordinate is zero.
• Similarly, \( \cos(n\pi) \) produces -1 or 1, depending on whether \( n \) is odd or even.

These simplifications are incredibly useful in calculations and problem-solving because they reduce complex expressions to elementary arithmetic. So, remembering that \( \sin(n\pi) = 0 \) can help quickly evaluate trigonometric sums like the one in the exercise.

Evaluating Trigonometric Expressions

When faced with trigonometric expressions, especially those involving sums, breaking them into smaller parts can make the evaluation process easier. With expressions that include integer multiples of \( \pi \), certain patterns and properties of trigonometric functions facilitate quick evaluation.

• First, recognize the periodicity and specific values of trigonometric functions at these multiples. For instance, \( \sin(n\pi) = 0 \).
• In the provided exercise, each term of the sum \( \sin(k\pi) \) is zero for \( k = 1\) to 5, which drastically simplifies the sum to zero.
• Understanding these principals means you can evaluate complex-looking expressions rapidly without exhaustive calculations.

By observing the properties of trigonometric functions and their behavior at specific angles, you can develop strategies to swiftly assess and conclude on various trigonometric expressions.