Question For the rotation $$ \frac{60 \pi}{7} $$, find the coterminal angle from $$ 0 \leq heta

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Question For the rotation $$ \frac{60 \pi}{7} $$, find the coterminal angle from $$ 0 \leq 	heta<2 \pi $$, the quadrant, and the reference angle. Answer Attempt s out of a The coterminal angle is

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To solve the problem of finding a coterminal angle for $$ \frac{60\pi}{7} $$ radians within the interval $$ 0 \leq \theta < 2\pi $$, as well as determining its quadrant and reference angle, follow these steps:

### 1. **Finding the Coterminal Angle**

A coterminal angle is obtained by adding or subtracting multiples of $$ 2\pi $$ from the given angle until the result lies within the desired interval $$ [0, 2\pi) $$.

Given:
$$
\theta = \frac{60\pi}{7}
$$

To find a coterminal angle within $$ [0, 2\pi) $$, subtract multiples of $$ 2\pi $$:
$$
\theta_{\text{coterminal}} = \frac{60\pi}{7} - k \cdot 2\pi
$$
where $$ k $$ is the integer number of full rotations to subtract.

Calculate the number of full rotations ($$ k $$):
$$
k = \left\lfloor \frac{\frac{60\pi}{7}}{2\pi} \right\rfloor = \left\lfloor \frac{60}{14} \right\rfloor = 4
$$

Subtract $$ 4 $$ full rotations:
$$
\theta_{\text{coterminal}} = \frac{60\pi}{7} - 4 \cdot 2\pi = \frac{60\pi}{7} - \frac{56\pi}{7} = \frac{4\pi}{7}
$$

**Coterminal Angle:** $$ \theta = \frac{4\pi}{7} $$ radians

### 2. **Determining the Quadrant**

To identify the quadrant in which $$ \theta = \frac{4\pi}{7} $$ lies, compare it with the standard angle boundaries:

- **Quadrant I:** $$ 0 \leq \theta < \frac{\pi}{2} $$
- **Quadrant II:** $$ \frac{\pi}{2} \leq \theta < \pi $$
- **Quadrant III:** $$ \pi \leq \theta < \frac{3\pi}{2} $$
- **Quadrant IV:** $$ \frac{3\pi}{2} \leq \theta < 2\pi $$

Calculate:
$$
\frac{4\pi}{7} \approx 1.795 \text{ radians}
$$
$$
\frac{\pi}{2} \approx 1.571 \text{ radians}, \quad \pi \approx 3.142 \text{ radians}
$$

Since $$ 1.571 < 1.795 < 3.142 $$, the angle $$ \frac{4\pi}{7} $$ lies in **Quadrant II**.

### 3. **Finding the Reference Angle**

The reference angle is the smallest acute angle that the terminal side of $$ \theta $$ makes with the x-axis.

For an angle in **Quadrant II**:
$$
\text{Reference Angle} = \pi - \theta = \pi - \frac{4\pi}{7} = \frac{3\pi}{7}
$$

**Reference Angle:** $$ \frac{3\pi}{7} $$ radians

### **Summary**

- **Coterminal Angle:** $$ \frac{4\pi}{7} $$ radians
- **Quadrant:** II
- **Reference Angle:** $$ \frac{3\pi}{7} $$ radians
Coterminal Angle: $$ \frac{4\pi}{7} $$ radians Quadrant: II Reference Angle: $$ \frac{3\pi}{7} $$ radians

Question For the rotation \( \frac{60 \pi}{7} \), find the coterminal angle from \( 0 \leq \theta<2 \pi \), the quadrant, and the reference angle. Answer Attempt s out of a The coterminal angle is

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