Evaluate -(3pi)/4+pi/4
The question is asking to perform a calculation with trigonometric constants, π (pi). Specifically, it requires the evaluation of an expression that subtracts a fraction of π from another fraction of π. The overall objective is to simplify the expression by combining the two terms, which both involve π, and express the result in terms of π or another simplified form if possible.
$- \frac{3 \pi}{4} + \frac{\pi}{4}$
Answer
Solution:
Step 1:
Combine the terms with the same denominator to simplify the expression. $\frac{-3\pi + \pi}{4}$
Step 2:
Simplify the numerator by adding $-3\pi$ and $\pi$. $\frac{-2\pi}{4}$
Step 3:
Reduce the fraction by eliminating common factors.
Step 3.1:
Extract the factor of $2$ from the numerator. $\frac{2(-\pi)}{4}$
Step 3.2:
Identify and remove any common factors between the numerator and denominator.
Step 3.2.1:
Factor out a $2$ from the denominator. $\frac{2(-\pi)}{2(2)}$
Step 3.2.2:
Eliminate the common factor of $2$. $\frac{\cancel{2}(-\pi)}{\cancel{2}\cdot 2}$
Step 3.2.3:
Rewrite the simplified expression. $\frac{-\pi}{2}$
Step 4:
Position the negative sign in front of the fraction for the final answer. $-\frac{\pi}{2}$
Step 5:
Present the final result in various formats.
Exact Form: $-\frac{\pi}{2}$
Decimal Form: Approximately $-1.57079632679...$
Knowledge Notes:
Simplifying expressions with pi: When dealing with expressions that include the mathematical constant $\pi$, it's important to remember that $\pi$ is an irrational number and should be kept in symbolic form unless a decimal approximation is necessary.
Combining like terms: When you have terms with the same denominator, you can combine them by adding or subtracting the numerators.
Reducing fractions: To simplify a fraction, you can divide both the numerator and the denominator by their greatest common factor (GCF). In this case, both the numerator and the denominator are divisible by $2$.
Negative signs in fractions: The negative sign can be placed in front of the fraction, in the numerator, or in the denominator without changing the value of the fraction.
Decimal representation of pi: While $\pi$ is often used in its symbolic form, it can also be approximated as a decimal. The value of $\pi$ to several decimal places is approximately $3.14159265359...$, so half of $\pi$ would be approximately $1.57079632679...$.
Exact vs. Decimal Form: The exact form of an expression with $\pi$ is preferred in mathematical and scientific contexts for precision, while the decimal form can be used for practical calculations where an approximation is sufficient.
