1. Three Precalculus Problems to Check Answers

3/18/09

I have done three problems and need verification of my answers.

1. Write the rectangular equation x=3 in polar form. My answer: rcostheta=3

2. Simplify {(sq. root of 3) + i}^4 My answer: 8 + 8(sq.root3i

3. Find (costheta+isintheta)^2 My answer: cos^2theta - isin^2theta

2. Originally Posted by Joanie
3/18/09

I have done three problems and need verification of my answers.

1. Write the rectangular equation x=3 in polar form. My answer: rcostheta=3

2. Simplify {(sq. root of 3) + i}^4 My answer: 8 + 8(sq.root3i

3. Find (costheta+isintheta)^2 My answer: cos^2theta - isin^2theta

The first two are correct.

For the third

$\displaystyle (\cos{\theta} + i\sin{\theta})^2 = \cos^2{\theta} + 2i\cos{\theta}\sin{\theta} - \sin^2{\theta}$.

Remember that

$\displaystyle \cos^2{\theta} - \sin^2{\theta} = \cos{2\theta}$ and $\displaystyle 2\cos{\theta}\sin{\theta} = \sin{2\theta}$.

So $\displaystyle (\cos{\theta} + i\sin{\theta})^2 = \cos{2\theta} + i\sin{2\theta}$.