To find the reference angle for $\frac{7\pi}{9}$, follow these steps:

Step 1: Identify the Quadrant

Since $\frac{7\pi}{9}$ is measured in radians, let's first determine its approximate location on the unit circle.

• Convert $\frac{7\pi}{9}$ to degrees: $\frac{7\pi}{9} \times \frac{180^\circ}{\pi} = \frac{7 \times 180^\circ}{9} = 140^\circ$ This angle is in the second quadrant because it lies between $90^\circ$ and $180^\circ$.

Step 2: Find the Reference Angle

In the second quadrant, the reference angle is found by subtracting the given angle from $180^\circ$.

Thus: $180^\circ - 140^\circ = 40^\circ$

Now, convert $40^\circ$ back to radians: $40^\circ \times \frac{\pi}{180^\circ} = \frac{40\pi}{180} = \frac{2\pi}{9}$

Final Answer:

The reference angle for $\frac{7\pi}{9}$ is $\frac{2\pi}{9}$ radians.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find reference angles in different quadrants?
2. What is the reference angle for $\frac{8\pi}{9}$?
3. Can reference angles ever be negative?
4. How does the unit circle relate to reference angles?
5. What is the reference angle for an angle greater than $2\pi$?

Tip: Always remember that the reference angle is the smallest angle between the terminal side of the given angle and the x-axis.