Trigonometry - Trigonometric Graphs and Parameters

1. Consider the equation \(4x^2 + kx +1 = 0\). For what values of \(k\) does this equation have two equal roots?

Let \(f\) be the function \(f(\theta) = 2\cos{2\theta} + 4\cos{\theta} + 3\), for \( -360^{\circ} \leq \theta \leq 360^{\circ}\).

2. Show that this function may be written as \(f(\theta) = 4\cos^2{\theta} + 4\cos{\theta} + 1\).
3. Consider the equation \(f(\theta) = 0\), for \( -360^{\circ} \leq \theta \leq 360^{\circ}\).
   1. How many distinct values of \(\cos{\theta}\) satisfy this equation?
   2. Find all values of \(\theta\) which satisfy this equation.
4. Given that \(f(\theta) = c\) is satisfied by only three values of \(\theta\), find the value of \(c\).