Chapter 8

Approximate the area of triangle \(A B C\). $$a=25.0, \quad b=80.0, \quad c=60.0$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 3+2 i $$

Exer. \(35-40:\) Prove the property if a and \(b\) are vectors and \(m\) is a real number. $$ (\mathbf{a}+\mathbf{b}) \cdot(\mathbf{a}-\mathbf{b})=\mathbf{a} \cdot \mathbf{a}-\mathbf{b} \cdot \mathbf{b} $$

Approximate the area of triangle \(A B C\). $$a=20.0, \quad b=20.0, \quad c=10.0$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -3+i $$

A triangular field has sides of lengths \(a, b\), and \(c\) (in yards). Approximate the number of acres in the field \(\left(1\right.\) acre \(\left.=4840 \mathrm{yd}^{2}\right)\) $$a=115, \quad b=140, \quad c=200$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -4+2 i $$

A triangular field has sides of lengths \(a, b\), and \(c\) (in yards). Approximate the number of acres in the field \(\left(1\right.\) acre \(\left.=4840 \mathrm{yd}^{2}\right)\) $$a=320, \quad b=350, \quad c=500$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -5-3 i $$

Approximate the area of a parallelogram that has sides of lengths \(a\) and \(b\) (in feet) if one angle at a vertex has measure \(\boldsymbol{\theta}\). $$a=12.0, \quad b=16.0, \quad \theta=40^{\circ}$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ -2-7 i $$

Approximate the area of a parallelogram that has sides of lengths \(a\) and \(b\) (in feet) if one angle at a vertex has measure \(\boldsymbol{\theta}\). $$a=40.3, \quad b=52.6, \quad \theta=100^{\circ}$$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 4-3 i $$

Exer. 45-46: Vectors are used extensively in computer graphics to perform shading. When light strikes a flat surface, it is reflected, and that area should not be shaded. Suppose that an incoming ray of light is represented by a vector \(L\) and that \(N\) is a vector orthogonal to the flat surface, as shown in the figure. The ray of reflected light can be represented by the vector \(R\) and is calculated using the formula \(\mathbf{R}=\mathbf{2}(\mathbf{N} \cdot \mathbf{L}) \mathbf{N}-\mathrm{L}\). Compute \(\mathbf{R}\) for the vectors \(L\) and \(N\). $$ \mathbf{L}=\left\langle-\frac{4}{5}, \frac{3}{5}\right\rangle, \quad \mathbf{N}=\langle 0,1\rangle $$

Exer. 21-46: Express the complex number in trigonometric form with \(0 \leq \theta<2 \pi\). $$ 1-3 i $$

Exer. 45-46: Vectors are used extensively in computer graphics to perform shading. When light strikes a flat surface, it is reflected, and that area should not be shaded. Suppose that an incoming ray of light is represented by a vector \(L\) and that \(N\) is a vector orthogonal to the flat surface, as shown in the figure. The ray of reflected light can be represented by the vector \(R\) and is calculated using the formula \(\mathbf{R}=\mathbf{2}(\mathbf{N} \cdot \mathbf{L}) \mathbf{N}-\mathrm{L}\). Compute \(\mathbf{R}\) for the vectors \(L\) and \(N\). $$ \mathbf{L}=\left\langle\frac{12}{13},-\frac{5}{13}\right\rangle, \quad \mathbf{N}=\left\langle\frac{1}{2} \sqrt{2}, \frac{1}{2} \sqrt{2}\right\rangle $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 4\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 8\left(\cos \frac{7 \pi}{4}+i \sin \frac{7 \pi}{4}\right) $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 6\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right) $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 12\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right) $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 5(\cos \pi+i \sin \pi) $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ 3\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right) $$

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{34} \operatorname{cis}\left(\tan ^{-1} \frac{3}{5}\right) $$

Find a vector that has the same direction as \(\langle-6,3\rangle\) and (a) twice the magnitude (b) one-half the magnitude

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{53} \operatorname{cis}\left[\tan ^{-1}\left(-\frac{2}{7}\right)\right] $$

Find a vector that has the opposite direction of \(8 \mathbf{i}-5 \mathbf{j}\) and (a) three times the magnitude (b) one-third the magnitude

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{5} \operatorname{cis}\left[\tan ^{-1}\left(-\frac{1}{2}\right)\right] $$

Find a vector of magnitude 6 that has the opposite direction of \(\mathbf{a}=4 \mathbf{i}-7 \mathbf{j}\).

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{10} \operatorname{cis}\left(\tan ^{-1} 3\right) $$

Find a vector of magnitude 4 that has the opposite direction of \(a=\langle 2,-5\rangle\).

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=-1+i, \quad z_{2}=1+i $$

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=\sqrt{3}-i, \quad z_{2}=-\sqrt{3}-i $$

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=-2-2 \sqrt{3} i, \quad z_{2}=5 i $$

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=-5+5 i, \quad z_{2}=-3 i $$

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=2 i, \quad z_{2}=-3 i $$

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=7, \quad z_{2}=3+5 i $$

Airplane course and ground speed Refer to Exercise 63. An airplane is flying in the direction \(140^{\circ}\) with an airspeed of \(500 \mathrm{mi} / \mathrm{hr}\), and a \(30 \mathrm{mi} / \mathrm{hr}\) wind is blowing in the direction \(65^{\circ}\). Approximate the true course and ground speed of the airplane.

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=-5, \quad z_{2}=3-2 i $$

Airplane course and ground speed An airplane pilot wishes to maintain a true course in the direction \(250^{\circ}\) with a ground speed of \(400 \mathrm{mi} / \mathrm{hr}\) when the wind is blowing directly north at \(50 \mathrm{mi} / \mathrm{hr}\). Approximate the required airspeed and compass heading.

Exer. 57-66: Use trigonometric forms to find \(z_{1} z_{2}\) and \(z_{1} / z_{2}\). $$ z_{1}=-3, \quad z_{2}=5+2 i $$

Wind direction and speed An airplane is flying in the direction \(20^{\circ}\) with an airspeed of \(300 \mathrm{mi} / \mathrm{hr}\). Its ground speed and true course are \(350 \mathrm{mi} / \mathrm{hr}\) and \(30^{\circ}\), respectively. Approximate the direction and speed of the wind.

Rowboat navigation The current in a river flows directly from the west at a rate of \(1.5 \mathrm{ft} / \mathrm{sec}\). A person who rows a boat at a rate of \(4 \mathrm{ft} / \mathrm{sec}\) in still water wishes to row directly north across the river. Approximate, to the nearest degree, the direction in which the person should row.

Exer. 69-72: The trigonometric form of complex numbers is often used by electrical engineers to describe the current \(I\), voltage \(V\), and impedance \(Z\) in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115 -volt, alternating current. The relationship among these three quantities is \(I=V / Z\). Approximate the unknown quantity, and express the answer in rectangular form to two decimal places. Finding voltage \(I=10 \operatorname{cis} 35^{\circ}, \quad Z=3 \operatorname{cis} 20^{\circ}\)

Exer. 69-72: The trigonometric form of complex numbers is often used by electrical engineers to describe the current \(I\), voltage \(V\), and impedance \(Z\) in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115 -volt, alternating current. The relationship among these three quantities is \(I=V / Z\). Approximate the unknown quantity, and express the answer in rectangular form to two decimal places. Finding voltage \(\quad I=12 \operatorname{cis} 5^{\circ}, \quad Z=100 \operatorname{cis} 90^{\circ}\)

Exer. 69-72: The trigonometric form of complex numbers is often used by electrical engineers to describe the current \(I\), voltage \(V\), and impedance \(Z\) in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115 -volt, alternating current. The relationship among these three quantities is \(I=V / Z\). Approximate the unknown quantity, and express the answer in rectangular form to two decimal places. Finding impedance \(I=8\) cis \(5^{\circ}, \quad V=115\) cis \(45^{\circ}\)

Exer. 69-72: The trigonometric form of complex numbers is often used by electrical engineers to describe the current \(I\), voltage \(V\), and impedance \(Z\) in electrical circuits with alternating current. Impedance is the opposition to the flow of current in a circuit. Most common electrical devices operate on 115 -volt, alternating current. The relationship among these three quantities is \(I=V / Z\). Approximate the unknown quantity, and express the answer in rectangular form to two decimal places. Finding current \(\quad Z=78 \operatorname{cis} 61^{\circ}, \quad V=163 \operatorname{cis} 17^{\circ}\)

The modulus of the impedance \(Z\) represents the total opposition to the flow of electricity in a circuit and is measured in ohms. If \(Z=14-13 i\), compute \(|Z|\).

The real part of \(V\) represents the actual voltage delivered to an electrical appliance in volts. Approximate this voltage when \(I=4\) cis \(90^{\circ}\) and \(Z=18\) cis \(\left(-78^{\circ}\right)\).

The real part of \(I\) represents the actual current delivered to an electrical appliance in amps. Approximate this current when \(V=163\) cis \(43^{\circ}\) and \(Z=100\) cis \(17^{\circ}\).