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.

$- \frac{3 π}{4} + \frac{π}{4}$

Answer

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Solution:

Step 1:

Combine the terms with the same denominator to simplify the expression. $\frac{- 3 π + π}{4}$

Step 2:

Simplify the numerator by adding $- 3 π$ and $π$ . $\frac{- 2 π}{4}$

Step 3:

Reduce the fraction by eliminating common factors.

Step 3.1:

Extract the factor of $2$ from the numerator. $\frac{2 (- π)}{4}$

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. $\frac{2 (- π)}{2 (2)}$

Step 3.2.2:

Eliminate the common factor of $2$ . $\frac{2 (- π)}{2 \cdot 2}$

Step 3.2.3:

Rewrite the simplified expression. $\frac{- π}{2}$

Step 4:

Position the negative sign in front of the fraction for the final answer. $- \frac{π}{2}$

Step 5:

Present the final result in various formats.

Exact Form: $- \frac{π}{2}$

Decimal Form: Approximately $- 1.57079632679 . . .$

Knowledge Notes:

Simplifying expressions with pi: When dealing with expressions that include the mathematical constant $π$ , it's important to remember that $π$ 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 $π$ is often used in its symbolic form, it can also be approximated as a decimal. The value of $π$ to several decimal places is approximately $3.14159265359 . . .$ , so half of $π$ would be approximately $1.57079632679 . . .$ .
Exact vs. Decimal Form: The exact form of an expression with $π$ is preferred in mathematical and scientific contexts for precision, while the decimal form can be used for practical calculations where an approximation is sufficient.