To express $\cos(3\theta)$ as a polynomial in $\cos(\theta)$, we can use the triple angle formula for cosine. The formula is given by:

$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$

Derivation:

1. Using the angle addition formula: $\cos(3\theta) = \cos(2\theta + \theta)$ We can apply the cosine addition formula: $\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$ Setting $A = 2\theta$ and $B = \theta$: $\cos(3\theta) = \cos(2\theta)\cos(\theta) - \sin(2\theta)\sin(\theta)$

2. Finding $\cos(2\theta)$ and $\sin(2\theta)$: $\cos(2\theta) = 2\cos^2(\theta) - 1$ $\sin(2\theta) = 2\sin(\theta)\cos(\theta) = 2\sqrt{1 - \cos^2(\theta)}\cos(\theta)$

3. Substituting back into the formula: $\cos(3\theta) = (2\cos^2(\theta) - 1)\cos(\theta) - 2\sqrt{1 - \cos^2(\theta)}\cos(\theta)\sin(\theta)$

   However, for the polynomial form, we can simplify the cosine part: $\cos(3\theta) = 2\cos^3(\theta) - \cos(\theta) - 2\sin(\theta)(1 - \cos^2(\theta))$

4. Final simplification leads to: $\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$

Conclusion:

Thus, the polynomial representation of $\cos(3\theta)$ in terms of $\cos(\theta)$ is:

$\cos(3\theta) = 4\cos^3(\theta) - 3\cos(\theta)$

Would you like more details on the derivation or any other related concepts? Here are some questions you might find interesting:

1. What are other angle formulas for cosine?
2. How does the triple angle formula relate to the unit circle?
3. Can you derive similar formulas for sine?
4. How can these formulas be applied in trigonometric equations?
5. What is the significance of polynomials in trigonometry?

Tip: When working with trigonometric identities, always consider using known formulas to simplify expressions efficiently.