State Euler's formula.

State Euler's identity obtained from Euler's formula when \(\theta=\pi\).

Use Euler's formula to write \(e^{-i\theta}\) in terms of sine and cosine.

Evaluate \(e^{i0}\) using Euler's formula.

Evaluate \(e^{i\pi/3}\) in Cartesian form.

Evaluate \(e^{-i\pi/4}\) in Cartesian form.

Derive \(\cos\theta=(e^{i\theta}+e^{-i\theta})/2\).

Derive \(\sin\theta=(e^{i\theta}-e^{-i\theta})/(2i)\).

Use Euler's formula to compute \(e^{i\pi/6}e^{i\pi/3}\).

Use Euler's formula to show that \(|e^{i\theta}|=1\).

Explain how Euler's formula connects complex exponentials to circular motion.

A student writes \(e^{i\theta}=\cos\theta+\sin\theta\). Identify the error.

Use Euler's formula to simplify \(e^{i\theta}+e^{i(\theta+\pi)}\).

Use Euler's formula to express \(2\cos\theta\) as a sum of complex exponentials.

For which real \(\theta\in[0,2\pi)\) is \(e^{i\theta}=i\)?

For which real \(\theta\in[0,2\pi)\) is \(e^{i\theta}\) real?

For which real \(\theta\) does \(e^{i\theta}=e^{-i\theta}\)?

Use Taylor series to justify Euler's formula by grouping real and imaginary terms.

Prove \(e^{i(\theta+2\pi)}=e^{i\theta}\) using Euler's formula.

A learner argues that \(e^{i\pi}=e^i e^\pi\). Explain why this is incorrect and give the correct value.