Chapter 2: Complex Numbers

Verify the results given for the roots in Example 4. You can find the exact values in terms of $\sqrt{\mathbf{3}}$ by using trigonometric addition formulas or more easily by using a computer to solve ${\mathit{z}}^{\mathbf{6}}\mathbf{=}\mathbf{-}\mathbf{8}\mathit{i}$. (You still may have to do a little work by hand to put the computer’s solution into the given form.)

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

${\mathbf{\left(}\frac{\mathbf{1}\mathbf{+}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{i}}\mathbf{\right)}}^{\mathbf{5}}$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.$\mathbf{\left|}\mathbf{4}{\mathbf{e}}^{\mathbf{2}\mathbf{i}\mathbf{-}\mathbf{1}}\mathbf{\right|}$

Find each of the following in the x + iyform and check your answers by computer.

$\mathit{s}\mathit{i}\mathit{n}\frac{\mathbf{i}\mathbf{\pi }}{\mathbf{2}}$

Solve for all possible values of the real numbers $\mathit{x}$ and $\mathit{y}$ in the following equations.

$\mathit{x}\mathbf{+}\mathit{i}\mathit{y}\mathbf{=}\mathbf{3}\mathit{i}\mathbf{-}\mathbf{4}$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\mathbf{|}\mathbf{3}{\mathbf{e}}^{\mathbf{5}\mathbf{i}}\mathbf{\times }\mathbf{7}{\mathbf{e}}^{\mathbf{-}\mathbf{2}\mathbf{i}}\mathbf{|}$

Find each of the following in the x + iyform and check your answers by computer.

$\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{h}\mathbf{(}\mathit{i}\mathit{\pi }\mathbf{+}\mathbf{2}\mathbf{)}$

Solve for all possible values of the real numbers xand y in the following equations.

$\mathbf{2}\mathit{i}\mathit{x}\mathbf{+}\mathbf{3}\mathbf{=}\mathit{y}\mathbf{-}\mathit{i}$

Use Problems 27 and 28 to find the following absolute values. If you understand Problems 27 and 28 and equation (5.1), you should be able to do these in your head.

$\mathbf{|}\mathbf{2}{\mathbf{e}}^{\mathbf{i\pi }\mathbf{/}\mathbf{6}}{\mathbf{|}}^{\mathbf{2}}$

Find each of the following in the x + iyform and check your answers by computer.

$\mathit{s}\mathit{i}\mathit{n}\mathit{h}\left(1+\frac{i\pi }{2}\right)$