From the image, it looks like the question asks you to compute the sine and cosine of $\frac{5\pi}{6}$ using the reference angle.

Let me guide you step by step:

1. Find the Reference Angle:

   • The angle $\frac{5\pi}{6}$ is in the second quadrant because $\frac{5\pi}{6}$ lies between $\frac{\pi}{2}$ and $\pi$.
   • The reference angle is calculated as $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.

2. In Which Quadrant?:

   • As determined, $\frac{5\pi}{6}$ is in the second quadrant. Therefore, the quadrant is 2.

3. Compute the Sine and Cosine:

   • In the second quadrant:
     • Sine is positive.
     • Cosine is negative.
   • Using the reference angle $\frac{\pi}{6}$, the sine and cosine values are derived from known trigonometric values:
     • $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$, and since sine is positive in the second quadrant: $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$
     • $\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$, and since cosine is negative in the second quadrant: $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

Correct Answers:

• Reference Angle: $\frac{\pi}{6}$
• Quadrant: 2
• $\sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}$
• $\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2}$

Would you like further clarification or additional examples?

5 Questions for Deeper Understanding:

1. How do you determine the reference angle for angles in the third and fourth quadrants?
2. Why are the sine and cosine signs different in each quadrant?
3. How do you compute other trigonometric values like tangent or cotangent for similar angles?
4. Can you generalize the reference angle for angles larger than $2\pi$?
5. What happens if the given angle is negative—how does that affect the quadrant and reference angle?

Tip:

Always visualize the unit circle when solving trigonometric problems. It helps to determine the quadrant and sign of each trigonometric function!