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.
$\frac{\pi}{2} - \frac{\pi}{3}$
Answer
Solution:
Step 1:
To find a common denominator for $\frac{\pi}{2}$, multiply it by $\frac{3}{3}$ to get $\frac{\pi}{2} \cdot \frac{3}{3} - \frac{\pi}{3}$.
Step 2:
Similarly, for $-\frac{\pi}{3}$, multiply it by $\frac{2}{2}$ to achieve a common denominator: $\frac{\pi}{2} \cdot \frac{3}{3} - \frac{\pi}{3} \cdot \frac{2}{2}$.
Step 3:
Establish a common denominator of $6$ for both fractions.
Step 3.1:
Multiply $\frac{\pi}{2}$ by $\frac{3}{3}$: $\frac{\pi \cdot 3}{2 \cdot 3} - \frac{\pi}{3} \cdot \frac{2}{2}$.
Step 3.2:
Calculate $2 \cdot 3$: $\frac{\pi \cdot 3}{6} - \frac{\pi}{3} \cdot \frac{2}{2}$.
Step 3.3:
Multiply $\frac{\pi}{3}$ by $\frac{2}{2}$: $\frac{\pi \cdot 3}{6} - \frac{\pi \cdot 2}{3 \cdot 2}$.
Step 3.4:
Calculate $3 \cdot 2$: $\frac{\pi \cdot 3}{6} - \frac{\pi \cdot 2}{6}$.
Step 4:
Combine the numerators over the common denominator of $6$: $\frac{\pi \cdot 3 - \pi \cdot 2}{6}$.
Step 5:
Simplify the numerator.
Step 5.1:
Rearrange to place the coefficient before $\pi$: $\frac{3 \cdot \pi - \pi \cdot 2}{6}$.
Step 5.2:
Multiply $2$ by $-1$: $\frac{3 \pi - 2 \pi}{6}$.
Step 5.3:
Subtract $2 \pi$ from $3 \pi$: $\frac{\pi}{6}$.
Step 6:
Present the result in various formats.
Exact Form: $\frac{\pi}{6}$
Decimal Form: $0.52359877 \ldots$
Knowledge Notes:
To solve the problem of finding the exact value of $\frac{\pi}{2} - \frac{\pi}{3}$, 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, $3\pi - 2\pi$ simplifies to $\pi$.
Pi ($\pi$): 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 $\frac{\pi}{6}$, we can also express it as a decimal. However, since $\pi$ is irrational, any decimal representation will be an approximation.
