The equation you're working with is:

$\cos \theta = \frac{1}{2}$

General Solution:

To find the general solution for this equation, we first recognize that the cosine function has the value $\frac{1}{2}$ at specific angles in the unit circle.

• The angle $\theta = \frac{\pi}{3}$ (or 60°) satisfies the equation, since $\cos \left( \frac{\pi}{3} \right) = \frac{1}{2}$.
• Additionally, cosine is positive in both the first and fourth quadrants. Therefore, the other solution in the range of one full rotation (0 to $2\pi$) will be $\theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}$.

Thus, the general solutions are:

$\theta = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \theta = \frac{5\pi}{3} + 2n\pi$

Where $n \in \mathbb{Z}$, indicating that the solutions repeat every full revolution of the unit circle (multiples of $2\pi$).

Final General Solution:

$\theta = \frac{\pi}{3} + 2n\pi \quad \text{or} \quad \theta = \frac{5\pi}{3} + 2n\pi, \quad n \in \mathbb{Z}$

Would you like further clarification or have other related questions?

Here are some related questions to explore:

1. What are the solutions for other trigonometric functions like sine or tangent?
2. How do periodicity and symmetry affect the general solution of trigonometric equations?
3. How would the solution change if $\cos \theta = -\frac{1}{2}$?
4. How do you graph the general solution of $\cos \theta = \frac{1}{2}$?
5. How would you use the unit circle to solve trigonometric equations?

Tip: Remember, for periodic functions like cosine, adding multiples of the period (in this case, $2\pi$) gives you the general solution.