$sin\frac{31\pi}{3}$ is equal to

Q: $sin\frac{31\pi}{3}$ is equal to

To solve the problem of finding the value of $ \sin\frac{31\pi}{3} $, we need to simplify the angle first. 1. **Simplification of the Angle:** \[ \frac{31\pi}{3} = 10\pi + \frac{\pi}{3} \] The angle $ 10\pi $ represents 5 full rotations (since $ 2\pi $ is one full rotation), and thus it does not affect the sine value. Therefore, we only need to consider the angle $ \frac{\pi}{3} $. 2. **Finding the Sine Value:** \[ \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \] This is a well-known trigonometric value for the angle $ \frac{\pi}{3} $. 3. **Verification of the Correct Option:** - Option 3: $ \frac{\sqrt{3}}{2} $ is indeed the correct value for $ \sin\left(\frac{\pi}{3}\right) $. 4. **Explanation of Incorrect Options:** - Option 1: $ \frac{1}{2} $ is the sine value for $ \frac{\pi}{6} $, not $ \frac{\pi}{3} $. - Option 2: $ \sqrt{3} $ is not a valid sine value for any standard angle in the unit circle. - Option 4: 0 is the sine value for angles like 0, $ \pi $, $ 2\pi $, etc., not for $ \frac{\pi}{3} $. In conclusion, the correct answer is Option 3: $ \frac{\sqrt{3}}{2} $, as it accurately represents the sine of the angle $ \frac{\pi}{3} $.

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Accepted Answer

To solve the problem of finding the value of $ \sin\frac{31\pi}{3} $, we need to simplify the angle first.

1. **Simplification of the Angle:**
   $$
   \frac{31\pi}{3} = 10\pi + \frac{\pi}{3}
   $$
   The angle $ 10\pi $ represents 5 full rotations (since $ 2\pi $ is one full rotation), and thus it does not affect the sine value. Therefore, we only need to consider the angle $ \frac{\pi}{3} $.

2. **Finding the Sine Value:**
   $$
   \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
   $$
   This is a well-known trigonometric value for the angle $ \frac{\pi}{3} $.

3. **Verification of the Correct Option:**
   - Option 3: $ \frac{\sqrt{3}}{2} $ is indeed the correct value for $ \sin\left(\frac{\pi}{3}\right) $.

4. **Explanation of Incorrect Options:**
   - Option 1: $ \frac{1}{2} $ is the sine value for $ \frac{\pi}{6} $, not $ \frac{\pi}{3} $.
   - Option 2: $ \sqrt{3} $ is not a valid sine value for any standard angle in the unit circle.
   - Option 4: 0 is the sine value for angles like 0, $ \pi $, $ 2\pi $, etc., not for $ \frac{\pi}{3} $.

In conclusion, the correct answer is Option 3: $ \frac{\sqrt{3}}{2} $, as it accurately represents the sine of the angle $ \frac{\pi}{3} $.