The image contains two questions asking for the reference angles of given angles in radians:

1. The reference angle of $\frac{12\pi}{7}$ radians.
2. The reference angle of $\frac{35\pi}{8}$ radians.

To solve these:

Step 1: Understanding Reference Angles

A reference angle is the smallest angle made between the terminal side of an angle and the x-axis. It's always positive and lies between $0$ and $\frac{\pi}{2}$ radians (or between $0^\circ$ and $90^\circ$).

Step 2: Convert the angles into their equivalent within $[0, 2\pi]$ range.

1. For $\frac{12\pi}{7}$:

• $\frac{12\pi}{7}$ radians is greater than $2\pi$ (since $2\pi = \frac{14\pi}{7}$).
• To find the coterminal angle, subtract $2\pi$ from $\frac{12\pi}{7}$: $\frac{12\pi}{7} - 2\pi = \frac{12\pi}{7} - \frac{14\pi}{7} = -\frac{2\pi}{7}$
• Since the angle is negative, we add $2\pi$ to get it positive: $-\frac{2\pi}{7} + 2\pi = \frac{12\pi}{7}$ This angle is now positive within the first quadrant. So, the reference angle is $\frac{2\pi}{7}$ radians.

2. For $\frac{35\pi}{8}$:

• $\frac{35\pi}{8}$ radians is greater than $2\pi$ (since $2\pi = \frac{16\pi}{8}$).
• To find the coterminal angle, subtract $2\pi$ multiple times: $\frac{35\pi}{8} - 2\pi = \frac{35\pi}{8} - \frac{16\pi}{8} = \frac{19\pi}{8}$
• Since this is still greater than $2\pi$, subtract $2\pi$ again: $\frac{19\pi}{8} - \frac{16\pi}{8} = \frac{3\pi}{8}$
• Now $\frac{3\pi}{8}$ lies within $[0, \pi]$, so it is the reference angle.

Final Answers:

1. The reference angle for $\frac{12\pi}{7}$ is $\frac{2\pi}{7}$ radians.
2. The reference angle for $\frac{35\pi}{8}$ is $\frac{3\pi}{8}$ radians.

Would you like further explanation on how to find reference angles?

Here are 5 related questions to explore this concept further:

1. How do reference angles differ between different quadrants?
2. How do you calculate the reference angle for angles in degrees?
3. What is the reference angle of $\frac{9\pi}{4}$?
4. How do reference angles help in solving trigonometric functions?
5. How do coterminal angles relate to reference angles?

Tip: When dealing with angles larger than $2\pi$ (or 360°), always reduce them by subtracting multiples of $2\pi$.