The Correct Answer and Explanation is:
To find the exact value of sin(5π4)\sin \left( \frac{5\pi}{4} \right)sin(45π), we can follow these steps:
- Identify the angle’s position on the unit circle:
The angle 5π4\frac{5\pi}{4}45π is in radians. To better understand its position, we can convert it to degrees. 5π4×180∘π=225∘.\frac{5\pi}{4} \times \frac{180^\circ}{\pi} = 225^\circ.45π×π180∘=225∘. The angle 225∘225^\circ225∘ lies in the third quadrant of the unit circle, where the sine function is negative. - Find the reference angle:
The reference angle is the acute angle that the terminal side of 225∘225^\circ225∘ makes with the x-axis. To find this, subtract 180∘180^\circ180∘ from 225∘225^\circ225∘: 225∘−180∘=45∘.225^\circ – 180^\circ = 45^\circ.225∘−180∘=45∘. Thus, the reference angle is 45∘45^\circ45∘ (or π4\frac{\pi}{4}4π). - Determine the sine value:
The sine of 45∘45^\circ45∘ or π4\frac{\pi}{4}4π is known to be 22\frac{\sqrt{2}}{2}22. Since the angle 225∘225^\circ225∘ is in the third quadrant, the sine function is negative in this quadrant. Therefore: sin(225∘)=−22.\sin \left( 225^\circ \right) = -\frac{\sqrt{2}}{2}.sin(225∘)=−22. - Conclusion:
The exact value of sin(5π4)\sin \left( \frac{5\pi}{4} \right)sin(45π) is: sin(5π4)=−22.\sin \left( \frac{5\pi}{4} \right) = -\frac{\sqrt{2}}{2}.sin(45π)=−22.
