- Understand the period of \(\sin \theta\) and its values in the third and fourth quadrants.
- Check each angle: \(\frac{5\pi}{4}\), \(\frac{7\pi}{4}\), and \(\frac{13\pi}{4}\) satisfy \(\sin \theta = -\frac{\sqrt{2}}{2}\).
- Discard \(\frac{3\pi}{4}\) and \(\frac{9\pi}{4}\) as they yield \(\frac{\sqrt{2}}{2}\).
- The final set of angles is \(\boxed{\frac{5\pi}{4}, \frac{7\pi}{4}, \frac{13\pi}{4}}\).
Explanation
- Analyzing the Problem and Trigonometric Properties
We are given a set of five angles \(\left\{\frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}, \frac{9\pi}{4}, \frac{13\pi}{4}\right\}\) and asked to identify which ones satisfy the trigonometric equation \(\sin \theta = -\frac{\sqrt{2}}{2}.\)
The sine function, \(\sin \theta\), is periodic with a period of \(2\pi\), meaning \(\sin(\theta + 2k\pi) = \sin \theta\) for any integer \(k.\) We know that \(\sin \theta = -\frac{\sqrt{2}}{2}\) occurs when the angle \(\theta\) lies in the third or fourth quadrant of the unit circle, with a reference angle of \(\frac{\pi}{4}.\) Specifically, the solutions in the interval \([0, 2\pi)\) are:
- Third quadrant: \(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\)
- Fourth quadrant: \(\theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4}\)
The general solution is \(\theta = \frac{5\pi}{4} + 2k\pi\) or \(\theta = \frac{7\pi}{4} + 2k\pi\) for any integer \(k.\)
- Evaluating the Given Angles
We now evaluate each of the given angles to see if they satisfy the condition \(\sin \theta = -\frac{\sqrt{2}}{2}.\)
- For \(\theta = \frac{3\pi}{4}\): \(\sin\left(\frac{3\pi}{4}\right) = \frac{\sqrt{2}}{2}.\) This does not satisfy the equation.
- For \(\theta = \frac{5\pi}{4}\): This is one of our fundamental solutions. \(\sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}.\) This satisfies the equation.
- For \(\theta = \frac{7\pi}{4}\): This is one of our fundamental solutions. \(\sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2}.\) This satisfies the equation.
- For \(\theta = \frac{9\pi}{4}\): Subtract the period \(2\pi = \frac{8\pi}{4}\) to find the coterminal angle: \(\frac{9\pi}{4} - \frac{8\pi}{4} = \frac{\pi}{4}.\) Thus, \(\sin\left(\frac{9\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}.\) This does not satisfy the equation.
- For \(\theta = \frac{13\pi}{4}\): Subtract the period \(2\pi = \frac{8\pi}{4}\) to find the coterminal angle: \(\frac{13\pi}{4} - \frac{8\pi}{4} = \frac{5\pi}{4}.\) Thus, \(\sin\left(\frac{13\pi}{4}\right) = \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}.\) This satisfies the equation.
- Conclusion
Based on the evaluation above, the angles that satisfy \(\sin \theta = -\frac{\sqrt{2}}{2}\) are \(\frac{5\pi}{4}\), \(\frac{7\pi}{4}\), and \(\frac{13\pi}{4}\). The final answer is \(\boxed{\frac{5\pi}{4}, \frac{7\pi}{4}, \frac{13\pi}{4}}.\)
Examples
In navigation and aviation, pilots often use circular coordinates to determine their heading relative to a base direction. Just as finding the correct angle for a specific sine value helps identify the correct orientation on a unit circle, understanding these periodic relationships allows navigators to calculate correct paths even after making multiple full rotations or completing loops in their journey. This ensures that regardless of how many times a pilot circles a waypoint, they can always identify their precise direction based on standardized trigonometric values.