Pregunta
10) Evaluate: \( \sin \left(\frac{2 \pi}{3}\right) \)
Ask by Mathis Powell. in the United States
Feb 26,2025
Solución de inteligencia artificial de Upstudy
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sin(2π/3) = √3/2
Solución
To evaluate sin(2π/3), we can follow these steps:
1. Recognize that 2π/3 is in the second quadrant, where sine is positive.
2. The reference angle for 2π/3 is π/3. That is, 2π/3 = π - π/3.
3. Since sin(π - θ) = sin(θ), we have sin(2π/3) = sin(π/3).
4. We know that sin(π/3) = √3/2.
Therefore, sin(2π/3) = √3/2.
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Simplifique esta solución Extra Insights
To evaluate \( \sin \left(\frac{2 \pi}{3}\right) \), we can recognize that \( \frac{2 \pi}{3} \) radians is located in the second quadrant of the unit circle. The reference angle is \( \pi - \frac{2 \pi}{3} = \frac{\pi}{3} \). The sine function is positive in the second quadrant, so we can use the sine of the reference angle: \[ \sin \left(\frac{2 \pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}. \] Thus, \[ \sin \left(\frac{2 \pi}{3}\right) = \frac{\sqrt{3}}{2}. \]
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