To find the reference angle, we need to understand the position of the given angle in relation to the nearest x-axis and then find the acute angle that it makes with the x-axis. Here's how to handle each case:

(a) $\frac{9\pi}{10}$

1. This is already in radians. Since it's less than $\pi$ (which is $\frac{10\pi}{10}$), it's located in the second quadrant.
2. To find the reference angle for angles in the second quadrant, we subtract the given angle from $\pi$. $\text{Reference Angle} = \pi - \frac{9\pi}{10} = \frac{10\pi}{10} - \frac{9\pi}{10} = \frac{\pi}{10}$ So, the reference angle is $\frac{\pi}{10}$.

(b) $\frac{11\pi}{8}$

1. Since $\frac{11\pi}{8} > \pi$, this angle is in the third quadrant. For third quadrant angles, the reference angle is found by subtracting $\pi$ from the given angle. $\text{Reference Angle} = \frac{11\pi}{8} - \pi = \frac{11\pi}{8} - \frac{8\pi}{8} = \frac{3\pi}{8}$ Thus, the reference angle is $\frac{3\pi}{8}$.

(c) $\frac{10\pi}{3}$

1. Since $\frac{10\pi}{3}$ is greater than $2\pi$, we subtract $2\pi$ (i.e., $\frac{6\pi}{3}$) to bring it within one full revolution. $\frac{10\pi}{3} - 2\pi = \frac{10\pi}{3} - \frac{6\pi}{3} = \frac{4\pi}{3}$
2. $\frac{4\pi}{3}$ is in the third quadrant. For third quadrant angles, we subtract $\pi$ from the angle: $\text{Reference Angle} = \frac{4\pi}{3} - \pi = \frac{4\pi}{3} - \frac{3\pi}{3} = \frac{\pi}{3}$ Thus, the reference angle is $\frac{\pi}{3}$.

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Summary:

• (a) $\frac{9\pi}{10}$: Reference angle is $\frac{\pi}{10}$.
• (b) $\frac{11\pi}{8}$: Reference angle is $\frac{3\pi}{8}$.
• (c) $\frac{10\pi}{3}$: Reference angle is $\frac{\pi}{3}$.

Would you like further clarification on any of these? Here are five questions that extend the concept of reference angles:

1. How would you find the reference angle for $\frac{13\pi}{4}$?
2. What is the reference angle for $-\frac{7\pi}{6}$?
3. Can you explain why reference angles are always positive and acute?
4. How do reference angles relate to the unit circle?
5. How does the reference angle change for angles in the fourth quadrant?

Tip: When calculating reference angles, always reduce angles greater than $2\pi$ by subtracting $2\pi$ to find the equivalent angle in the first full rotation.