Chapter 7.7, Problem 111E

To determine

(a)

To say the statement ${\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}\right)}^n=\cos \frac{n\pi }{6}+i\sin \frac{n\pi }{6}$ is true or false.

To determine

(b)

To say the statement ${\left(\sin \frac{\pi }{3}+i\cos \frac{\pi }{3}\right)}^n=\sin \frac{n\pi }{3}+i\cos \frac{n\pi }{3}$ is true or false.

To determine

(c)

To say the statement ${\left(\cos \left(-\frac{\pi }{3}\right)+i\sin \left(-\frac{\pi }{3}\right)\right)}^n=\cos \frac{n\pi }{3}-i\sin \frac{n\pi }{3}$ is true or false.

To determine

(d)

To say the statement ${\left(\cos \frac{\pi }{6}+i\sin \frac{\pi }{3}\right)}^n=\cos \frac{n\pi }{6}+i\sin \frac{n\pi }{3}$ is true or false.

To determine

(e)

To say the statement ${\left(\cos \theta +i\sin \theta \right)}^n=\cos n\theta +i\sin n\theta$ is true or false.

To determine

(f)

To say the statement ${\left(\cos \theta -i\sin \theta \right)}^n=\cos n\theta -i\sin n\theta$ is true or false.

To determine

(g)

To say the statement ${\left(\cos +i\sin \theta \right)}^{-n}=\cos \theta -i\sin \theta$ is true or false.

To determine

(h)

To say the statement ${\left(\cos \theta -i\sin \theta \right)}^{-n}=\cos n\theta +i\sin n\theta$ is true or false.

i.

To determine

To say the statement ${\left(\sin \theta +i\cos \theta \right)}^n=\sin n\theta +i\cos n\theta$ is true or false.

j.

To determine

To say the statement ${\left(\sin \theta +i\cos \theta \right)}^n=\cos \left[n\left(\frac{\pi }{2}-\theta \right)\right]+i\sin \left[n\left(\frac{\pi }{2}-\theta \right)\right]$ is true or false.