Use a reference angle to write sin(31π/36) in terms of the sine of a positive acute angle. Include parentheses in your answer.

We need to express \( \sin\left(\frac{31\pi}{36}\right) \) in terms of the sine of a positive acute angle using a reference angle.

1. **Step 1: Identify the quadrant.**

   First, simplify the given angle:
   \[
   \frac{31\pi}{36} = \frac{31}{36} \pi
   \]
   This angle is between \( \pi \) and \( 2\pi \) because \( \pi = \frac{36\pi}{36} \) and \( 2\pi = \frac{72\pi}{36} \), meaning:
   \[
   \pi < \frac{31\pi}{36} < 2\pi
   \]
   Therefore, the angle is in the third quadrant.

2. **Step 2: Find the reference angle.**

   The reference angle \( \theta_{\text{ref}} \) is the difference between the given angle and \( \pi \) (since we are in the third quadrant):
   \[
   \theta_{\text{ref}} = \frac{31\pi}{36} - \pi = \frac{31\pi}{36} - \frac{36\pi}{36} = \frac{31\pi - 36\pi}{36} = \frac{-5\pi}{36}
   \]
   The reference angle is \( \frac{5\pi}{36} \).

3. **Step 3: Determine the sine function in the third quadrant.**

   In the third quadrant, sine is negative. Therefore:
   \[
   \sin\left(\frac{31\pi}{36}\right) = -\sin\left(\frac{5\pi}{36}\right)
   \]

Thus, the final expression is:
\[
\sin\left(\frac{31\pi}{36}\right) = -\sin\left(\frac{5\pi}{36}\right)
\]

Would you like more details on any of the steps, or do you have any questions?

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Here are 5 related questions:
1. How do you find the reference angle for any angle in radians?
2. Why is the sine function negative in the third quadrant?
3. Can you express \( \sin(7\pi/6) \) using a reference angle?
4. What is the cosine of an angle in the third quadrant?
5. How do the sine and cosine functions behave in each quadrant?

**Tip:** Always check the quadrant first when working with trigonometric functions to determine the sign.

Use a reference angle to write sin(31π/36) in terms of the sine of a positive acute angle. Include parentheses in your answer.

Learn how to express sin(31π/36) in terms of the sine of a positive acute angle by using reference angles. This step-by-step guide covers trigonometric properties of sine in different quadrants and the reference angle formula. Suitable for students in Grades 10-12.

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