Questions: Determine the coterminal angle of the following angle. Your answer should be in the interval (0 leq theta<2 pi). Write your answer in reduced form; if your answer is (12 pi / 13), write (12 pi / 13). (theta = 14 pi / 3)

Determine the coterminal angle of the following angle. Your answer should be in the interval (0 leq theta<2 pi). Write your answer in reduced form; if your answer is (12 pi / 13), write (12 pi / 13).

(theta = 14 pi / 3)

Transcript text: 0:05:28 elapsed Question 4 (1 point) 3 Determine the coterminal angle of the following angle. Your answer should be in the interval $0 \leq \theta<2 \pi$. Write your answer in reduced form; if your answer is $\frac{12 \pi}{13}$, write $12 \mathrm{pi} / 13$. 9 \[ \theta=\frac{14 \pi}{3} \] $\square$ A

Solution

Solution Steps

Step 1: Identify the given angle

The given angle is $\theta = \frac{14\pi}{3}$.

Step 2: Determine the coterminal angle within the interval $0 \leq \theta < 2\pi$

To find a coterminal angle within the interval $0 \leq \theta < 2\pi$, subtract multiples of $2\pi$ from the given angle until the result falls within the desired interval.

\[ \frac{14\pi}{3} - 2\pi \cdot k \]

Step 3: Calculate the value of $k$

First, determine how many full rotations ($2\pi$) are contained in $\frac{14\pi}{3}$:

\[ k = \left\lfloor \frac{\frac{14\pi}{3}}{2\pi} \right\rfloor = \left\lfloor \frac{14}{6} \right\rfloor = 2 \]

Subtract $2 \cdot 2\pi$ from $\frac{14\pi}{3}$:

\[ \frac{14\pi}{3} - 2 \cdot 2\pi = \frac{14\pi}{3} - \frac{12\pi}{3} = \frac{2\pi}{3} \]

The coterminal angle within the interval $0 \leq \theta < 2\pi$ is $\frac{2\pi}{3}$.