Find $\tan 20° - 33\tan^3 20° + 27\tan^2 20° - 4 = $
Step-by-Step Solution
Key Concept: The triple angle formula $\tan 3\alpha = \frac{3\tan\alpha - \tan^3\alpha}{1 - 3\tan^2\alpha}$ relates $\tan 60°$ to $\tan 20°$.
Setting $2\theta = k$, we use the triple angle formula $\tan 3\alpha = \frac{3\tan\alpha - \tan^3\alpha}{1 - 3\tan^2\alpha}$. With $\alpha = 20°$, we get $\tan 60° = \frac{3\tan 20° - \tan^3 20°}{1 - 3\tan^2 20°}$. Since $\tan 60° = \sqrt{3}$, this leads to a cubic equation in $\tan 20°$: $k^3 - 33k^2 + 27k - 3 = 0$ where $k = \tan 20°$. Solving the cubic equation yields $\tan 20° = 3\tan 20° - 4\tan^3 20° = 7$.
Correct Answer: 7