Find the reference angle for a rotation of start fraction, 7, pi, divided by, 9, end fraction 97π​ .

1. Break Down the Problem

To find the reference angle for a given angle in radians, we follow these steps:

1. Simplify the angle to its equivalent within the range $[0, 2\pi]$.
2. Determine the reference angle based on the quadrant in which the angle lies.

2. Relevant Concepts

• The reference angle is the acute angle formed by the terminal side of the given angle and the x-axis.
• For angles in different quadrants:
  • Quadrant I: Reference angle $\theta$.
  • Quadrant II: Reference angle $\pi - \theta$.
  • Quadrant III: Reference angle $\theta - \pi$.
  • Quadrant IV: Reference angle $2\pi - \theta$.

3. Analysis and Detail

1. Simplifying the Angle: Given angle: $\theta = 97\pi$ We need to find the equivalent angle in the range $[0, 2\pi]$. To do this, we can calculate: $97\pi \mod 2\pi$ This is equivalent to calculating: $97 \mod 2 = 1$ Thus, the equivalent angle is: $\theta_{equiv} = 1\pi = \pi$

2. Determining the Reference Angle: The angle $\pi$ lies on the negative x-axis, which means it is in Quadrant II. The reference angle is calculated as: $\text{Reference angle} = \pi - \theta_{equiv} = \pi - \pi = 0$

4. Verify and Summarize

We verified that $97\pi$ is equivalent to $\pi$ when simplified into the $[0, 2\pi]$ range. The reference angle calculation also holds true, as $\pi$ corresponds to an angle of $0$ with the x-axis.

Final Answer

The reference angle for a rotation of $97\pi$ is $0$.