Complete each problem below by hand. Show your work and provide a reason for each step. Calculators are not allowed on this assignment.
There are many key theorems involving sin and cos. You may have heard of Euler’s Theorem which states: $e^{i\theta}=\cos\theta+i\sin\theta$.
- Determine the equation for Euler’s Theorem when $\theta=\pi$. This is called Euler’s Identity. In your own words, without the use of outside resources, make a conjuncture about why this identity is often described as the most beautiful equation.
- Another key theorem is de Moivre’s Theorem which states: $$(\cos\theta+i\sin\theta)^n=\cos n\theta+i\sin n\theta$$
Using this theorem to prove the identity, $\cos{3\theta}=4\cos^3{\theta}−3\cos{\theta}$. Let $n=3$.
This identity is of particular importance in geometry as it plays a key part in proving what angles can be constructed using a compass and straightedge.
- Verify the identity $\frac{\cos(−x)}{1−\sin(x)}+\frac{1+\sin(−x)}{cos(x)}=2\sec x$ Show all work and justify your steps.
