Chapter 7

(a) use the theorem on page 590 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Cube roots of -125

(a) use the theorem on page 590 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Fifth roots of \(128(-1+i)\)

(a) use the theorem on page 590 to find the indicated roots of the complex number, (b) represent each of the roots graphically, and (c) write each of the roots in standard form. Fifth roots of \(4(1-i)\)

Use the theorem on page 590 to find all the solutions of the equation, and represent the solutions graphically. $$x^{3}+1=0$$

The formula \(E=I Z,\) where \(E\) represents voltage, \(I\) represents current, and \(Z\) represents impedance (a measure of opposition to a sinusoidal electric current \(,\) is used in electrcal engineering. Each variable is a complex number. Use the formula to find the missing quantity for the given conditions. Then convert the given conditions to trigonometric form and check your result. $$\begin{aligned}&I=10+2 i\\\&Z=4+3 i\end{aligned}$$

The formula \(E=I Z,\) where \(E\) represents voltage, \(I\) represents current, and \(Z\) represents impedance (a measure of opposition to a sinusoidal electric current \(,\) is used in electrcal engineering. Each variable is a complex number. Use the formula to find the missing quantity for the given conditions. Then convert the given conditions to trigonometric form and check your result. $$\begin{aligned}&I=12+2 i\\\&Z=3+5 i\end{aligned}$$

The formula \(E=I Z,\) where \(E\) represents voltage, \(I\) represents current, and \(Z\) represents impedance (a measure of opposition to a sinusoidal electric current \(,\) is used in electrcal engineering. Each variable is a complex number. Use the formula to find the missing quantity for the given conditions. Then convert the given conditions to trigonometric form and check your result. $$\begin{aligned}&I=2+4 i\\\&E=5+5 i\end{aligned}$$

The formula \(E=I Z,\) where \(E\) represents voltage, \(I\) represents current, and \(Z\) represents impedance (a measure of opposition to a sinusoidal electric current \(,\) is used in electrcal engineering. Each variable is a complex number. Use the formula to find the missing quantity for the given conditions. Then convert the given conditions to trigonometric form and check your result. $$\begin{aligned}&E=12+24 i\\\&Z=12+20 i\end{aligned}$$

The formula \(E=I Z,\) where \(E\) represents voltage, \(I\) represents current, and \(Z\) represents impedance (a measure of opposition to a sinusoidal electric current \(,\) is used in electrcal engineering. Each variable is a complex number. Use the formula to find the missing quantity for the given conditions. Then convert the given conditions to trigonometric form and check your result. $$\begin{aligned}&E=15+12 i\\\&Z=25+24 i\end{aligned}$$

Determine whether the statement is true or false. Justify your answer. \(\frac{1}{2}(1-\sqrt{3} i)\) is a ninth root of \(-1\).

Determine whether the statement is true or false. Justify your answer. \(\sqrt{3}+i\) is a solution of the equation \(x^{2}-8 i=0\).

Determine whether the statement is true or false. Justify your answer. The product of two complex numbers is 0 only when the modulus of one (or both) of the complex numbers is 0.

Determine whether the statement is true or false. Justify your answer. Geometrically, the \(n\) th roots of any complex number \(z\) are equally spaced around the unit circle centered at the origin.

Given two complex numbers \(\quad z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right) \quad\) and \(z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right), z_{2} \neq 0,\) show that \(\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]\).

Show that \(\bar{z}=r[\cos (-\theta)+i \sin (-\theta)]\) is the complex conjugate of \(z=r(\cos \theta+i \sin \theta)\).

Show that the negative of \(z=r(\cos \theta+i \sin \theta)\) is \(-z=r[\cos (\theta+\pi)+i \sin (\theta+\pi)]\).

The famous formula $$e^{a+b i}=e^{a}(\cos b+i \sin b)$$ is called Euler's Formula, after the Swiss mathematician Leonhard Euler \((1707-1783) .\) This formula gives rise to the equation $$e^{\pi i}+1=0$$. This equation relates the five most famous numbers in mathematics \(-0,1, \pi, e,\) and \(i-\) in a single equation. Show how Euler's Formula can be used to derive this equation. Write a short paragraph summarizing your work.

For the simple harmonic motion described by the trigonometric function, find the maximum displacement from equilibrium and the lowest possible positive value of \(t\) for which \(d=0.\) $$d=16 \cos \frac{\pi}{4} t$$

For the simple harmonic motion described by the trigonometric function, find the maximum displacement from equilibrium and the lowest possible positive value of \(t\) for which \(d=0.\) $$d=\frac{1}{7} \sin \frac{5 \pi}{4} t$$

For the simple harmonic motion described by the trigonometric function, find the maximum displacement from equilibrium and the lowest possible positive value of \(t\) for which \(d=0.\) $$d=\frac{1}{8} \cos 12 \pi t$$

For the simple harmonic motion described by the trigonometric function, find the maximum displacement from equilibrium and the lowest possible positive value of \(t\) for which \(d=0.\) $$d=\frac{1}{12} \sin 60 \pi t$$