Math 361, Spring 2018, Assignment 11

Read:

1. Section 32.

Carefully define the following terms, then give one example and one non-example of each:

1. Geometric construction.
2. Constructible number.

Carefully state the following theorems (you do not need to prove them):

1. Quadratic formula.
2. Corollary of the quadratic formula concerning the roots of a quadratic polynomial ("If $p\in F[x]$ is quadratic and has roots in $E$ then the roots are contained in...")
3. Theorem relating constructible numbers to the coordinates of constructible points.
4. Theorem concerning sums, products, and inverses of constructible numbers ("The set $\mathcal{C}$ of constructible numbers is a...")

Solve the following problems:

1. Consult a reference on trigonometry (for instance here) and review the trigonometric identities that express $\sin(\alpha+\beta)$ and $\cos(\alpha+\beta)$ in terms of $\sin(\alpha), \cos(\alpha), \sin(\beta),$ and $\cos(\beta)$.
2. Using the identities you reviewed above, derive the double-angle and triple-angle identities for $\cos(2\theta)$ and $\cos(3\theta)$. (You can look up these identities here.)
3. Put $\theta=72^{\circ}$. Using a picture if necessary, explain why $\cos(2\theta)=\cos(3\theta)$.
4. Show that $\cos(72^{\circ})$ is a root of the polynomial $4x^3-2x^2-3x+1$.
5. Find a root of the above polynomial other than $\cos(72^{\circ})$.
6. Find an exact expression for $\cos(72^{\circ})$.
7. Devise a compass and straightedge construction that produces a regular pentagon.

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