The general solutions for \(\cos(2z) = \frac{\sqrt{3}}{2}\) are:
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- Question:** A biodiversity specialist is studying the periodic behavior of genetic expression modeled by the equation \(2\cos(2z) = \sqrt{3}\). Find the sum of all angles \(z \in [0^\circ, 360^\circ]\) that satisfy this equation.
- First, solve for \(\cos(2z)\):
- 2\cos(2z) = \sqrt{3} \quad \Rightarrow \quad \cos(2z) = \frac{\sqrt{3}}{2}
- 2z = 30^\circ + 360^\circ n \quad \text{or} \quad 2z = 330^\circ + 360^\circ n
- Divide by 2 to find \(z\):
- z = 15^\circ + 180^\circ n \quad \text{or} \quad z = 165^\circ + 180^\circ n
