Write \(\cos\theta\) in terms of complex exponentials.

Write \(\sin\theta\) in terms of complex exponentials.

Use the exponential formula to evaluate \(\cos 0\).

Use the exponential formula to evaluate \(\sin0\).

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

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

Use complex exponentials to show that \(\cos(-\theta)=\cos\theta\).

Use complex exponentials to show that \(\sin(-\theta)=-\sin\theta\).

Use complex exponentials to prove \(\cos^2\theta+\sin^2\theta=1\).

Use exponentials to derive \(\cos(\alpha+\beta)\).

Explain why the formulas for \(\cos\theta\) and \(\sin\theta\) from exponentials still produce real values for real \(\theta\).

A student writes \(\sin\theta=(e^{i\theta}-e^{-i\theta})/2\). Diagnose the error.

Use complex exponentials to express \(\cos^2\theta\) in terms of \(\cos 2\theta\).

Use complex exponentials to express \(\sin^2\theta\) in terms of \(\cos2\theta\).

For which real \(\theta\) does \(e^{i\theta}+e^{-i\theta}=0\)?

For which real \(\theta\) is \((e^{i\theta}-e^{-i\theta})/(2i)=1\)?

For which real \(\theta\) is \(e^{2i\theta}+e^{-2i\theta}=1\)?

Prove the angle addition formula for \(\sin(\alpha+\beta)\) using Euler's formula.

Prove \(\cosh^2 x-\sinh^2 x=1\) from the exponential definitions.

A learner says complex exponentials make trig identities unnecessary. Explain why the exponential forms are useful but do not remove the need to interpret trig identities.