Question 11.4.23

The triple angle formula states that \[\cos(3\theta) = 4\cos^3\theta - 3\cos\theta.\]

  1. We’re going to solve the cubic equation $x^3 = 3px + 2q$ for $x$ using the triple angle formula. If $x=2r\cos\theta$, find values of $r$ and $\theta$ that will solve the cubic equation. (In Question 7.5.34 we were able to solve this equation when $q^2-p^3 \ge 0$.) Show answer

  2. What values of $p$ and $q$ will your solution work for? Show hint

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  3. Using your solution, find all values of $x$ that satisfy the equation $x^3 = 6x + 4$. Show hint

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