Find the Exact Value pi/2-pi/3

The given problem asks to calculate the exact value of an expression involving the mathematical constants π (pi), which represents the ratio of a circle's circumference to its diameter. Specifically, the expression is a subtraction operation between two fractions: π/2 and π/3. To find the exact value, one would need to carry out the appropriate simplification of the subtraction by obtaining a common denominator, which in this case would be the least common multiple of 2 and 3, and then subtracting the numerators accordingly. The result would yield the exact value of the expression in terms of π.

$\frac{π}{2} - \frac{π}{3}$

Answer

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Solution:

Step 1:

To find a common denominator for $\frac{π}{2}$ , multiply it by $\frac{3}{3}$ to get $\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3}$ .

Step 2:

Similarly, for $- \frac{π}{3}$ , multiply it by $\frac{2}{2}$ to achieve a common denominator: $\frac{π}{2} \cdot \frac{3}{3} - \frac{π}{3} \cdot \frac{2}{2}$ .

Step 3:

Establish a common denominator of $6$ for both fractions.

Step 3.1:

Multiply $\frac{π}{2}$ by $\frac{3}{3}$ : $\frac{π \cdot 3}{2 \cdot 3} - \frac{π}{3} \cdot \frac{2}{2}$ .

Step 3.2:

Calculate $2 \cdot 3$ : $\frac{π \cdot 3}{6} - \frac{π}{3} \cdot \frac{2}{2}$ .

Step 3.3:

Multiply $\frac{π}{3}$ by $\frac{2}{2}$ : $\frac{π \cdot 3}{6} - \frac{π \cdot 2}{3 \cdot 2}$ .

Step 3.4:

Calculate $3 \cdot 2$ : $\frac{π \cdot 3}{6} - \frac{π \cdot 2}{6}$ .

Step 4:

Combine the numerators over the common denominator of $6$ : $\frac{π \cdot 3 - π \cdot 2}{6}$ .

Step 5:

Simplify the numerator.

Step 5.1:

Rearrange to place the coefficient before $π$ : $\frac{3 \cdot π - π \cdot 2}{6}$ .

Step 5.2:

Multiply $2$ by $- 1$ : $\frac{3 π - 2 π}{6}$ .

Step 5.3:

Subtract $2 π$ from $3 π$ : $\frac{π}{6}$ .

Step 6:

Present the result in various formats.

Exact Form: $\frac{π}{6}$

Decimal Form: $0.52359877 \dots$

Knowledge Notes:

To solve the problem of finding the exact value of $\frac{π}{2} - \frac{π}{3}$ , we need to understand the concept of common denominators in fractions. This is essential for adding, subtracting, or comparing fractions with different denominators.

Common Denominator: A common denominator is a shared multiple of the denominators of two or more fractions. In this case, the least common multiple (LCM) of $2$ and $3$ is $6$ . We use this to rewrite both fractions with the same denominator.
Multiplying by One: When we multiply a fraction by a form of one (like $\frac{3}{3}$ or $\frac{2}{2}$ ), we do not change its value. This technique is used to find equivalent fractions with a desired denominator.
Combining Fractions: Once fractions have a common denominator, their numerators can be combined (added or subtracted) while keeping the denominator the same.
Simplifying Fractions: After combining the numerators, the resulting fraction can often be simplified. In this case, $3 π - 2 π$ simplifies to $π$ .
Pi ( $π$ ): Pi is a mathematical constant approximately equal to $3.14159$ and is the ratio of a circle's circumference to its diameter. It is an irrational number, meaning it cannot be exactly expressed as a simple fraction and its decimal representation is infinite and non-repeating.
Decimal Representation: While the exact form of the answer is $\frac{π}{6}$ , we can also express it as a decimal. However, since $π$ is irrational, any decimal representation will be an approximation.