Verified Answer :
- The cosine function has a period of \(360^{\circ}\) and the general solutions for \(\cos \theta = \frac{1}{2}\) are \(\theta = \pm 60^{\circ} + 360^{\circ} k\).
- By testing the given angles, we find that \(-60^{\circ}\) is a primary solution.
- For \(660^{\circ}\), subtracting one period of \(360^{\circ}\) yields \(300^{\circ}\), which is equivalent to \(-60^{\circ} \pmod{360^{\circ}}\), so it is also a solution.
- The correct angles are \(\boxed{-60^{\circ}, 660^{\circ}}\).
Explanation
- Understanding the Problem
We are given the equation \(\cos \theta = \frac{1}{2}\) and a list of angle measures: \(-120^{\circ}\), \(-60^{\circ}\), \(120^{\circ}\), \(600^{\circ}\), and \(660^{\circ}\). Our goal is to identify which of these angles satisfy the equation. Recall that the cosine function is periodic with a period of \(360^{\circ}\), meaning \(\cos(\theta) = \cos(\theta + 360^{\circ} \cdot k)\) for any integer \(k\). Furthermore, the cosine function is even, meaning \(\cos(\theta) = \cos(-\theta)\). The fundamental solutions to \(\cos \theta = \frac{1}{2}\) in the interval \([0^{\circ}, 360^{\circ})\) are \(\theta = 60^{\circ}\) and \(\theta = 300^{\circ}\) (which is equivalent to \(-60^{\circ}\)). Therefore, the general solution for \(\cos \theta = \frac{1}{2}\) is \(\theta = \pm 60^{\circ} + 360^{\circ} \cdot k\) for any integer \(k\).
- Evaluating Candidate Angles
We test each candidate angle from the provided list \(\{-120^{\circ}, -60^{\circ}, 120^{\circ}, 600^{\circ}, 660^{\circ}\}\) against the general solution \(\theta = \pm 60^{\circ} + 360^{\circ} \cdot k\):
- For \(-120^{\circ}\): This does not match \(\pm 60^{\circ} + 360^{\circ} \cdot k\).
- For \(-60^{\circ}\): This matches \(-60^{\circ} + 360^{\circ} \cdot 0\). Thus, \(\cos(-60^{\circ}) = \frac{1}{2}\).
- For \(120^{\circ}\): This does not match \(\pm 60^{\circ} + 360^{\circ} \cdot k\).
- For \(600^{\circ}\): Subtracting \(360^{\circ}\), we get \(600^{\circ} - 360^{\circ} = 240^{\circ}\), which is not \(\pm 60^{\circ} + 360^{\circ} \cdot k\).
- For \(660^{\circ}\): Subtracting \(360^{\circ}\), we get \(660^{\circ} - 360^{\circ} = 300^{\circ}\), which is equivalent to \(-60^{\circ} + 360^{\circ} \cdot 1\). Thus, \(\cos(660^{\circ}) = \frac{1}{2}\).
- Conclusion
Based on the evaluation, the angles \(-60^{\circ}\) and \(660^{\circ}\) satisfy the equation \(\cos \theta = \frac{1}{2}\). The other angles in the set result in values other than \(\frac{1}{2}\). Therefore, the correct selections from the list are \(-60^{\circ}\) and \(660^{\circ}\). The final answer is \(\boxed{-60^{\circ}, 660^{\circ}}\).
Examples
When designing a circular gear system, engineers need to ensure that parts of the gear are aligned at specific angles to mesh properly. For example, if a contact point needs to occur at an angle where the projection on the x-axis (cosine) is half the total radius, finding all possible angular positions is essential for the mechanism to function correctly at various rotations. This mathematical approach helps in calculating those precise angles in periodic systems.