Question Which angle(s) is/are cotermineal with $\frac{5 \pi}{3}$ ? A) $-\frac{5 \pi}{3}$ B) $\frac{11 \pi}{3}$ C) $\frac{\pi}{3}$ D) All of the above

Studdy Solution

STEP 1

Assumptions
1. Two angles are coterminal if they differ by an integer multiple of $2\pi$ .
2. The given angle is $\frac{5\pi}{3}$ .
3. We need to check if each of the provided angles is coterminal with $\frac{5\pi}{3}$ .

STEP 2

First, let's define what it means for two angles to be coterminal. Two angles are coterminal if one angle can be obtained by adding or subtracting a full rotation (which is $2\pi$ radians) to the other angle.
$\text{Coterminal angle} = \text{Given angle} \pm k(2\pi)$
where $k$ is an integer.

STEP 3

Let's check if angle I, $-\frac{5\pi}{3}$ , is coterminal with $\frac{5\pi}{3}$ .
We will add $2\pi$ to $-\frac{5\pi}{3}$ to see if we can obtain $\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + 2\pi = -\frac{5\pi}{3} + \frac{6\pi}{3}$

STEP 4

Calculate the sum to see if it is equal to $\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + \frac{6\pi}{3} = \frac{1\pi}{3}$

STEP 5

Since $\frac{1\pi}{3}$ is not equal to $\frac{5\pi}{3}$ , we can try subtracting $2\pi$ from $\frac{5\pi}{3}$ to see if we get $-\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3}$

STEP 6

Calculate the difference to see if it is equal to $-\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{1\pi}{3}$

STEP 7

Since $-\frac{1\pi}{3}$ is not equal to $-\frac{5\pi}{3}$ , we can conclude that angle I is not coterminal with $\frac{5\pi}{3}$ by a single rotation. However, we should consider multiple rotations (i.e., $k > 1$ ) to see if they can be coterminal.

STEP 8

Let's try adding $2\pi$ multiple times to $-\frac{5\pi}{3}$ to find a coterminal angle.
$-\frac{5\pi}{3} + 2(2\pi) = -\frac{5\pi}{3} + \frac{12\pi}{3}$

STEP 9

Calculate the sum to see if it is equal to $\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + \frac{12\pi}{3} = \frac{7\pi}{3}$

STEP 10

Since $\frac{7\pi}{3}$ is not equal to $\frac{5\pi}{3}$ , let's subtract $2\pi$ multiple times from $\frac{5\pi}{3}$ to find a coterminal angle.
$\frac{5\pi}{3} - 2(2\pi) = \frac{5\pi}{3} - \frac{12\pi}{3}$

STEP 11

Calculate the difference to see if it is equal to $-\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - \frac{12\pi}{3} = -\frac{7\pi}{3}$

STEP 12

Since $-\frac{7\pi}{3}$ is not equal to $-\frac{5\pi}{3}$ , let's continue by adding another $2\pi$ to $-\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + 3(2\pi) = -\frac{5\pi}{3} + \frac{18\pi}{3}$

STEP 13

Calculate the sum to see if it is equal to $\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + \frac{18\pi}{3} = \frac{13\pi}{3}$

STEP 14

Since $\frac{13\pi}{3}$ is not equal to $\frac{5\pi}{3}$ , let's subtract another $2\pi$ from $\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - 3(2\pi) = \frac{5\pi}{3} - \frac{18\pi}{3}$

STEP 15

Calculate the difference to see if it is equal to $-\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - \frac{18\pi}{3} = -\frac{13\pi}{3}$

STEP 16

Since $-\frac{13\pi}{3}$ is not equal to $-\frac{5\pi}{3}$ , let's try adding $4(2\pi)$ to $-\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + 4(2\pi) = -\frac{5\pi}{3} + \frac{24\pi}{3}$

STEP 17

Calculate the sum to see if it is equal to $\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + \frac{24\pi}{3} = \frac{19\pi}{3}$

STEP 18

Since $\frac{19\pi}{3}$ is not equal to $\frac{5\pi}{3}$ , let's subtract $4(2\pi)$ from $\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - 4(2\pi) = \frac{5\pi}{3} - \frac{24\pi}{3}$

STEP 19

Calculate the difference to see if it is equal to $-\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - \frac{24\pi}{3} = -\frac{19\pi}{3}$

STEP 20

Since $-\frac{19\pi}{3}$ is not equal to $-\frac{5\pi}{3}$ , we can see that adding or subtracting multiples of $2\pi$ in this manner will not yield $-\frac{5\pi}{3}$ . However, we can observe that by adding $2\pi$ six times, we get:
$-\frac{5\pi}{3} + 6(2\pi) = -\frac{5\pi}{3} + \frac{36\pi}{3}$

STEP 21

Calculate the sum to see if it is equal to $\frac{5\pi}{3}$ .
$-\frac{5\pi}{3} + \frac{36\pi}{3} = \frac{31\pi}{3}$

STEP 22

Since $\frac{31\pi}{3}$ is not equal to $\frac{5\pi}{3}$ , we can subtract $2\pi$ six times from $\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - 6(2\pi) = \frac{5\pi}{3} - \frac{36\pi}{3}$

STEP 23

Calculate the difference to see if it is equal to $-\frac{5\pi}{3}$ .
$\frac{5\pi}{3} - \frac{36\pi}{3} = -\frac{31\pi}{3}$

STEP 24

Since $-\frac{31\pi}{3}$ is not equal to $-\frac{5\pi}{3}$ , we can conclude that no matter how many times we add or subtract $2\pi$ , we will not get $-\frac{5\pi}{3}$ as a coterminal angle with $\frac{5\pi}{3}$ .

STEP 25

Now let's check angle II, $\frac{11\pi}{3}$ , to see if it is coterminal with $\frac{5\pi}{3}$ .
We can subtract $2\pi$ from $\frac{11\pi}{3}$ to see if we can obtain $\frac{5\pi}{3}$ .
$\frac{11\pi}{3} - 2\pi = \frac{11\pi}{3} - \frac{6\pi}{3}$

STEP 26

Calculate the difference to see if it is equal to $\frac{5\pi}{3}$ .
$\frac{11\pi}{3} - \frac{6\pi}{3} = \frac{5\pi}{3}$

STEP 27

Since $\frac{5\pi}{3}$ is equal to $\frac{5\pi}{3}$ , we can conclude that angle II, $\frac{11\pi}{3}$ , is coterminal with $\frac{5\pi}{3}$ .

STEP 28

Finally, let's check angle III, $\frac{\pi}{3}$ , to see if it is coterminal with $\frac{5\pi}{3}$ .
We can add $2\pi$ to $\frac{\pi}{3}$ to see if we can obtain $\frac{5\pi}{3}$ .
$\frac{\pi}{3} + 2\pi = \frac{\pi}{3} + \frac{6\pi}{3}$

STEP 29

Calculate the sum to see if it is equal to $\frac{5\pi}{3}$ .
$\frac{\pi}{3} + \frac{6\pi}{3} = \frac{7\pi}{3}$

STEP 30

Since $\frac{7\pi}{3}$ is not equal to $\frac{5\pi}{3}$ , we can conclude that angle III, $\frac{\pi}{3}$ , is not coterminal with $\frac{5\pi}{3}$ .
The only angle that is coterminal with $\frac{5\pi}{3}$ is angle II, $\frac{11\pi}{3}$ . Therefore, the correct answer is:
B) II only

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