Question: Use $\cos(3θ) = 4\cos^3(θ)-3\cos(θ)$ to find the three roots of $8x^3-6x=1.$
Currently, I have figured out that if you double the triple angle formula above you can line the two equations up to get $\cos(3θ) = 1/2$. However, I don't know where to go from here.
Update:
After letting x = cos(θ), I have found that θ = 20 or θ = 100 as x = 1/2. However, the question states that there are three answers. How do I find the third answer?
Answer to updated question:
The third answer is -0.7660 as cos(420) = 1/2.
Hint: To find all three answers of $\cos 3 \theta = 1/2$, once we solve the equation, we get $\theta = 20^\circ + 120^\circ k$ (or $\theta = \pi/9 + 2\pi k/9$ in radians), so in order to find all three values, we would have to set $k = 0, 1 \text { and } 2$.