Solve the trigonometric equations: i) 4 3 theta = theta

Here are the solutions to the trigonometric equations:

i) $4 \cos^3 \theta = \cos \theta$

Step 1: Rearrange the equation and factor out $\cos \theta$.

$$4 \cos^3 \theta - \cos \theta = 0$$ $$\cos \theta (4 \cos^2 \theta - 1) = 0$$

Step 2: Set each factor to zero and solve for $\cos \theta$.

$$\cos \theta = 0 \quad \text{or} \quad 4 \cos^2 \theta - 1 = 0$$ $$4 \cos^2 \theta = 1$$ $$$$