Chapter 7

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$4(1-\sqrt{3} i)^{3}$$

Consider two forces $$\mathbf{F}_{1}=\langle 10,0\rangle \quad$$ and $$\quad \mathbf{F}_{2}=5\langle\cos \theta, \sin \theta\rangle$$ (a) Write \(\left\|\mathbf{F}_{1}+\mathbf{F}_{2}\right\|\) as a function of \(\theta\) (b) Use a graphing utility to graph the function for \(0 \leq \theta<2 \pi\) (c) Use the graph in part (b) to determine the range of the function. What is its maximum, and for what value of \(\theta\) does it occur? What is its minimum, and for what value of \(\theta\) does it occur? (d) Explain why the magnitude of the resultant is never 0.

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[5\left(\cos 140^{\circ}+i \sin 140^{\circ}\right)\right]^{3}$$

Give geometric descriptions of (a) vector addition and (b) scalar multiplication.

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[3\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)\right]^{4}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)^{10}$$

Prove that \((\cos \theta) \mathbf{i}+(\sin \theta) \mathbf{j}\) is a unit vector for any value of \(\bar{\theta}\).

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[2\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\right]^{12}$$

Write a program for a graphing utility that graphs two vectors and their difference given the vectors in component form.

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[2(\cos \pi+i \sin \pi)]^{14}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(\cos 0+i \sin 0)^{20}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[4\left(\cos 10^{\circ}+i \sin 10^{\circ}\right)\right]^{6}$$

It Identify each quantity as either a scalar or a vector. Explain. (a) muzzle velocity of a bullet (b) price of a company's stock (c) air temperature of a room (d) weight of an automobile (e) volume of a fish tank (f) current of a river

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[6\left(\cos 15^{\circ}+i \sin 15^{\circ}\right)\right]^{4}$$

Simplify the expression. $$\left(\frac{6 x^{4}}{7 y^{-2}}\right)\left(14 x^{-1} y^{5}\right)$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[3\left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right)\right]^{2}$$

Simplify the expression. $$\left(5 s^{5} t^{-5}\right)\left(\frac{3 s^{-2}}{50 t^{-1}}\right)$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$\left[2\left(\cos \frac{\pi}{10}+i \sin \frac{\pi}{10}\right)\right]^{5}$$

Simplify the expression. $$(18 x)^{0}(4 x y)^{2}\left(3 x^{-1}\right)$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(3-2 i)^{5}$$

Simplify the expression. $$\left(5 a b^{2}\right)\left(a^{-3} b^{0}\right)\left(2 a^{0} b\right)^{-2}$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$(\sqrt{5}-4 i)^{4}$$

Simplify the expression. $$\left(2.1 \times 10^{9}\right)\left(3.4 \times 10^{-4}\right)$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[2(\cos 1.25+i \sin 1.25)]^{4}$$

Simplify the expression. $$\left(6.5 \times 10^{-6}\right)\left(3.8 \times 10^{4}\right)$$

Use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[4(\cos 2.8+i \sin 2.8)]^{5}$$

Solve the equation.$$\cos x(\cos x+1)=0$$

Use DeMoivre's Theorem to verify the indicated root of the real number. \(-\frac{1}{2}(1+\sqrt{3} i)\) is a sixth root of 1.

Solve the equation.$$\sin x(2 \sin x+\sqrt{2})=0$$

Use DeMoivre's Theorem to verify the indicated root of the real number. \(2^{-1 / 4}(1-i)\) is a fourth root of \(-2\).

Solve the equation.$$3 \sec x+4=10$$

Find the square roots of the complex number. $$2 i$$

Solve the equation.$$7 \csc x+18=4$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(0<\theta<\pi / 2\).$$\sqrt{x^{2}+100}, x=10 \cot \theta$$

Find the square roots of the complex number. $$-3 i$$

Use the trigonometric substitution to write the algebraic expression as a trigonometric function of \(\theta,\) where \(0<\theta<\pi / 2\).$$\sqrt{x^{2}-49}, x=7 \sec \theta$$ .

Find the square roots of the complex number. $$-6 i$$

Find the square roots of the complex number. $$2-2 i$$

Find the square roots of the complex number. $$2+2 i$$

Find the square roots of the complex number. $$1+\sqrt{3} i$$

Find the square roots of the complex number. $$1-\sqrt{3} i$$

(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. Square roots of \(5\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\)

(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. Square roots of \(16\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\)

(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. Fourth roots of \(81\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)\)

(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 \(-25 i\)

(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. Fourth roots of \(625 i\)

(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. Cube roots of \(-\frac{125}{2}(1+\sqrt{3} i)\)

(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. Fourth roots of \(256 i\)

(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. Fourth roots of \(i\)

(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 \(1.\)