Find the sine and cosine of the given angle sin 5pi/6  and cos 5pi/6

To find \( \sin \left(\frac{5\pi}{6}\right) \) and \( \cos \left(\frac{5\pi}{6}\right) \), we can follow these steps:

### Step 1: Understanding the angle
The angle \( \frac{5\pi}{6} \) is in **radians**, which corresponds to \( 150^\circ \). This angle lies in the **second quadrant** of the unit circle, where:
- The sine of an angle is **positive**.
- The cosine of an angle is **negative**.

### Step 2: Reference angle
The reference angle for \( \frac{5\pi}{6} \) is:
\[
\theta_{\text{ref}} = \pi - \frac{5\pi}{6} = \frac{\pi}{6}
\]
This corresponds to \( 30^\circ \).

### Step 3: Using known values
We know that:
\[
\sin \left( \frac{\pi}{6} \right) = \frac{1}{2}
\]
\[
\cos \left( \frac{\pi}{6} \right) = \frac{\sqrt{3}}{2}
\]

### Step 4: Applying signs for the second quadrant
In the second quadrant:
- \( \sin \left( \frac{5\pi}{6} \right) \) is positive, so:
\[
\sin \left( \frac{5\pi}{6} \right) = \frac{1}{2}
\]
- \( \cos \left( \frac{5\pi}{6} \right) \) is negative, so:
\[
\cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}
\]

### Final Answer:
\[
\sin \left( \frac{5\pi}{6} \right) = \frac{1}{2}
\]
\[
\cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the tangent of \( \frac{5\pi}{6} \)?
2. How do you convert radians to degrees?
3. What is the sine of \( \frac{7\pi}{6} \)?
4. How do you find the reference angle for any angle?
5. How can you determine the sign of a trigonometric function in any quadrant?

**Tip**: Always remember the signs of trigonometric functions in each quadrant: All Students Take Calculus (All positive, Sine positive, Tangent positive, Cosine positive).

Find the sine and cosine of the given angle sin 5pi/6 and cos 5pi/6

Discover how to calculate sin(5π/6) and cos(5π/6) using the unit circle and reference angles. This problem covers essential trigonometry concepts such as sine and cosine in the second quadrant. Suitable for students in Grades 9-12.

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