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:
$e^{i\theta} = cos\theta + isin\theta$.
- 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:
$(cos\theta + isin\theta)^{n} = cosn\theta + isinn\theta$
Using this theorem to prove the identity, $cos3\theta=4cos^3\theta-3cos\theta$.
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
$\frac{cos(-x)}{1-sin(x)} + \frac{1+sin(-x)}{cos(x)} = 2sec x$
Show all work and justify your steps.
