- Determine the value of the reference angle \(\sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2}\).
- Use symmetry and quadrant signs to find \(\sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2}\), \(\sin(\frac{4\pi}{3}) = -\frac{\sqrt{3}}{2}\), and \(\sin(\frac{5\pi}{3}) = -\frac{\sqrt{3}}{2}\).
- Match the resulting sequence \(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}\) with the given options.
- The correct choice is \(\boxed{b}\).
Explanation
- Analyzing the Trigonometric Expressions
The objective is to find the values of \(\sin \left(\frac{\pi}{3}\right)\), \(\sin \left(\frac{2 \pi}{3}\right)\), \(\sin \left(\frac{4 \pi}{3}\right)\), and \(\sin \left(\frac{5 \pi}{3}\right)\). These angles are located in different quadrants on the unit circle. Recall that the sine function is positive in the first and second quadrants, and negative in the third and fourth quadrants. The reference angle for all these expressions is \(\frac{\pi}{3}\), and we know that \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\).
- Evaluating the Sine Values
We evaluate each sine function individually:
- \(\sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) (Quadrant I).
- For \(\sin \left(\frac{2 \pi}{3}\right)\), we use the identity \(\sin(\pi - \theta) = \sin(\theta)\): \(\sin \left(\pi - \frac{\pi}{3}\right) = \sin \left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\) (Quadrant II).
- For \(\sin \left(\frac{4 \pi}{3}\right)\), we use the identity \(\sin(\pi + \theta) = -\sin(\theta)\): \(\sin \left(\pi + \frac{\pi}{3}\right) = -\sin \left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\) (Quadrant III).
- For \(\sin \left(\frac{5 \pi}{3}\right)\), we use the identity \(\sin(2\pi - \theta) = -\sin(\theta)\): \(\sin \left(2\pi - \frac{\pi}{3}\right) = -\sin \left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}\) (Quadrant IV).
- Conclusion and Selection
The calculated sequence is \(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}\). Comparing this result with the given choices:
A\(\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}\)
B\(\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}\)
C\(\frac{1}{2}, -\frac{1}{2}, \frac{1}{2}, -\frac{1}{2}\)
D\(\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}, \frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}\)
The sequence matches choice b. The final answer is \(\boxed{b}\).
Examples
Trigonometric values are essential in many fields like engineering and physics. For example, when analyzing alternating current (AC) power in a circuit, the voltage follows a sinusoidal pattern. Understanding the values of sine at specific phase angles (like \(\frac{\pi}{3}\) or \(\frac{2\pi}{3}\)) allows engineers to calculate the instantaneous voltage, design better circuits, and ensure the efficiency of power transmission. Similar principles apply to studying sound waves, vibration analysis, and signal processing.