Question

question 4. points possible: 3
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a. represent the list of all integer multiples of 6 for all integers, n.
b. represent the list of all integer multiples of \\( \frac { 6 \cdot \pi } { 4 } \\) for all integers, n.
c. represent the list of all real numbers \\( \frac { 2 } { 3 } \\)-unit less than an integer multiple of \\( 2 \pi \\) for all integers, n.
2+2npi
-pi/2+2npi
b. represent the list of angle measures (in radians) for angles co - terminal with an angle measuring 2 radians. (use the variable n to denote the integer multiple of \\( 2 \pi \\) in your expression.)

Explanation:

Part a (Represent all integer multiples of 6)

Step 1: Define the form

An integer multiple of a number \( a \) is given by \( a \times n \), where \( n \) is an integer. For the number 6, the integer multiples will be \( 6n \) where \( n \in \mathbb{Z} \) ( \( \mathbb{Z} \) represents the set of all integers).

Step 1: Simplify the fraction

First, simplify \( \frac{6\pi}{4} \) to \( \frac{3\pi}{2} \).

Step 2: Define the multiple

An integer multiple of a number \( a \) is \( a \times n \) where \( n \) is an integer. So for \( \frac{3\pi}{2} \), the integer multiples are \( \frac{3\pi}{2}n \) where \( n \in \mathbb{Z} \).

Step 1: Define the integer multiple of \( 2\pi \)

An integer multiple of \( 2\pi \) is \( 2\pi n \) where \( n \in \mathbb{Z} \).

Step 2: Subtract \( \frac{2}{3} \)

We need to find a number that is \( \frac{2}{3} \) less than this multiple. So we subtract \( \frac{2}{3} \) from \( 2\pi n \).

Answer:

\( 6n \) for all integers \( n \)

Part b (Represent all integer multiples of \( \frac{6\pi}{4} \))