Problem 9 - Calculus Single Variable

Question

P In Exercises \(1-10\), write out the sum, and perform the addition. $$ \sum_{j=1}^{3} j \cdot \sin ^{2}(j \pi / 6) $$

Step-by-Step Solution

Answer

The sum is \( \frac{19}{4} \).

Step 1: Understand the Sigma Notation

The sigma notation \( \sum_{j=1}^{3} j \cdot \sin^2(j \pi / 6) \) indicates that we need to find the sum of the terms obtained by plugging in values from 1 to 3 for \( j \) in the expression \( j \cdot \sin^2(j \pi / 6) \).

Step 2: Calculate the First Term (j=1)

Plug \( j = 1 \) into the expression to get the first term: \( 1 \cdot \sin^2(1 \cdot \pi / 6) = 1 \cdot \sin^2(\pi / 6) \). Since \( \sin(\pi / 6) = \frac{1}{2} \), \( \sin^2(\pi / 6) = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \). Therefore, the first term is \( 1 \cdot \frac{1}{4} = \frac{1}{4} \).

Step 3: Calculate the Second Term (j=2)

Plug \( j = 2 \) into the expression to get the second term: \( 2 \cdot \sin^2(2 \cdot \pi / 6) = 2 \cdot \sin^2(\pi / 3) \). Since \( \sin(\pi / 3) = \frac{\sqrt{3}}{2} \), \( \sin^2(\pi / 3) = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \). Thus, the second term is \( 2 \cdot \frac{3}{4} = \frac{6}{4} = \frac{3}{2} \).

Step 4: Calculate the Third Term (j=3)

Plug \( j = 3 \) into the expression to get the third term: \( 3 \cdot \sin^2(3 \cdot \pi / 6) = 3 \cdot \sin^2(\pi / 2) \). Since \( \sin(\pi / 2) = 1 \), \( \sin^2(\pi / 2) = 1^2 = 1 \). Therefore, the third term is \( 3 \cdot 1 = 3 \).

Step 5: Add the Terms Together

Now, sum all the terms calculated in the previous steps: \( \frac{1}{4} + \frac{3}{2} + 3 \). To add these fractions together, find a common denominator. The common denominator for 4 and 2 is 4. Rewrite \( \frac{3}{2} \) as \( \frac{6}{4} \) and \( 3 \) as \( \frac{12}{4} \). Now sum the fractions: \( \frac{1}{4} + \frac{6}{4} + \frac{12}{4} = \frac{19}{4} \).

Key Concepts

Sum of Terms

When working with exercises involving sigma notation, understanding the concept of the sum of terms is crucial. In the given exercise, sigma notation is used to represent the summation of specific terms generated by an expression. This expression is repeated for each integer value starting from 1 up to a maximum value, which in this case, is 3.

Here's what happens step-by-step:

• First, identify the range of values for the variable (in this case, from 1 to 3 for \( j \)).
• Substitute each value of \( j \) into the expression \( j \cdot \sin^2(j \pi / 6) \).
• Calculate the resultant terms individually for each value of \( j \).

By following this method, you will compute a series of terms. Next, you simply add these terms together to get the total sum. This step-by-step approach helps simplify complex expressions and results in a manageable sum.

Trigonometric Functions

Trigonometric functions play a significant role in the given exercise, particularly the sine function. These functions are key in solving many problems in mathematics, particularly those involving periodic phenomena.

In the exercise, you are dealing with the sine function squared, denoted as \( \sin^2(x) \). This involves calculating the square of the sine of an angle. As you evaluate, consider these basic steps:

• Identify the angle provided — in this exercise, it is wrapped in the form \( j \cdot \pi / 6 \) for \( j = 1, 2, 3 \).
• Calculate \( \sin(x) \), while using known values for angles like \( \pi/6, \pi/3, \pi/2 \) which correspond to \( 30^\circ, 60^\circ, \) and \( 90^\circ \) respectively.
• Finding the square of these sine values yields \( \sin^2(x) \).

Understanding these trigonometric function calculations simplifies evaluating terms and contributes to finding the entire sum.

Common Denominator

Adding fractions can sometimes seem daunting, but understanding the concept of a common denominator makes it much easier. In this exercise, once we've calculated our individual terms, they need to be summed up.

Here’s how to use a common denominator effectively:

• Identify the denominators in the fractions you need to sum. In this case, the denominators are 4 and 2.
• Determine the least common denominator, which is a multiple of each original denominator. Here, it's 4.
• Convert each term to an equivalent fraction using the common denominator. For example, rewrite \( \frac{3}{2} \) as \( \frac{6}{4} \).
• Add these converted fractions together. Finally, the sum \( \frac{1}{4} + \frac{6}{4} + \frac{12}{4} \) equals \( \frac{19}{4} \).

By confidently finding and using a common denominator, you simplify the process of adding fractions and correctly solve the exercise.