Chapter 8

Find the two square roots of \(1+\sqrt{3} i\).

Exer. 13-16: Show that the vectors are parallel, and determine whether they have the same direction or opposite directions. $$ \mathbf{a}=3 \mathbf{i}-5 \mathbf{j}, \quad \mathbf{b}=-\frac{12}{7} \mathbf{i}+\frac{20}{7} \mathbf{j} $$

Solve \(\triangle A B C\). $$a=25.0, \quad b=80.0, \quad c=60.0$$

Exer. 11-20: Represent the complex number geometrically. $$ -2-6 i $$

Find the two square roots of \(-9 i\).

Exer. 13-16: Show that the vectors are parallel, and determine whether they have the same direction or opposite directions. $$ \mathbf{a}=-\frac{5}{2} \mathbf{i}+6 \mathbf{j}, \quad \mathbf{b}=-10 \mathbf{i}+24 \mathbf{j} $$

Solve \(\triangle A B C\). $$ a=20.0, \quad b=20.0, \quad c=10.0$$

Solve \(\triangle A B C\). $$\beta=113^{\circ} 10^{\prime}, \quad b=248, \quad c=195$$

Exer. 11-20: Represent the complex number geometrically. $$ -(3-6 i) $$

Find the four fourth roots of \(-1-\sqrt{3} i\).

Exer. 13-16: Show that the vectors are parallel, and determine whether they have the same direction or opposite directions. $$ \mathbf{a}=\left\langle\frac{2}{3}, \frac{1}{2}\right\rangle, \quad \mathbf{b}=\langle 8,6\rangle $$

Dimensions of a triangular plot The angle at one corner of a triangular plot of ground is \(73^{\circ} 40^{\prime}\), and the sides that meet at this corner are 175 feet and 150 feet long. Approximate the length of the third side.

Exer. 11-20: Represent the complex number geometrically. $$ (1+2 i)^{2} $$

Find the four fourth roots of \(-8+8 \sqrt{3} i\).

Exer. 13-16: Show that the vectors are parallel, and determine whether they have the same direction or opposite directions. $$ \mathbf{a}=\langle 6,18\rangle, \quad \mathbf{b}=\langle-4,-12\rangle $$

Surveying To find the distance between two points \(A\) and \(B\), a surveyor chooses a point \(C\) that is 420 yards from \(A\) and 540 yards from \(B\). If angle \(A C B\) has measure \(63^{\circ} 10^{\prime}\), approximate the distance between \(A\) and \(B\).

Solve \(\triangle A B C\). $$\gamma=73.01^{\circ}, \quad a=17.31, \quad c=20.24$$

Exer. 11-20: Represent the complex number geometrically. $$ 2 i(2+3 i) $$

Find the three cube roots of \(-27 i\).

Exer. 17-20: Determine \(m\) such that the two vectors are orthogonal. $$ 3 \mathbf{i}-2 \mathbf{j}, \quad 4 \mathbf{i}+5 m \mathbf{j} $$

Distance between automobiles Two automobiles leave a city at the same time and travel along straight highways that differ in direction by \(84^{\circ}\). If their speeds are \(60 \mathrm{mi} / \mathrm{hr}\) and \(45 \mathrm{mi} / \mathrm{hr}\), respectively, approximately how far apart are the cars at the end of 20 minutes?

Surveying To find the distance between two points \(A\) and \(B\) that lie on opposite banks of a river, a surveyor lays off a line segment \(A C\) of length 240 yards along one bank and determines that the measures of \(\angle B A C\) and \(\angle A C B\) are \(63^{\circ} 20^{\prime}\) and \(54^{\circ} 10^{\prime}\), respectively (see the figure). Approximate the distance between \(A\) and \(B\).

Exer. 11-20: Represent the complex number geometrically. $$ (-3 i)(2-i) $$

Find the three cube roots of \(64 i\).

$a+0=a $$

Exer. 17-20: Determine \(m\) such that the two vectors are orthogonal. $$ 4 m \mathbf{i}+\mathbf{j}, \quad 9 m \mathbf{i}-25 \mathbf{j} $$

Angles of a triangular plot A triangular plot of land has sides of lengths 420 feet, 350 feet, and 180 feet. Approximate the smallest angle between the sides.

Surveying To determine the distance between two points \(A\) and \(B\), a surveyor chooses a point \(C\) that is 375 yards from \(A\) and 530 yards from \(B\). If \(\angle B A C\) has measure \(49^{\circ} 30^{\prime}\), approximate the distance between \(A\) and \(B\).

Exer. 11-20: Represent the complex number geometrically. $$ (1+i)^{2} $$

Exer. 19-22: Find the indicated roots, and represent them geometrically. The six sixth roots of unity

Exer. 17-20: Determine \(m\) such that the two vectors are orthogonal. $$ 9 \mathbf{i}-16 m \mathbf{j}, \quad \mathbf{i}+4 m \mathbf{j} $$

\(\mathbf{a}+(-\mathbf{a})=0 \quad \)

Distance between ships A ship leaves port at 1:00 P.M. and travels \(\mathrm{S} 35^{\circ} \mathrm{E}\) at the rate of \(24 \mathrm{mi} / \mathrm{hr}\). Another ship leaves the same port at 1:30 P.M. and travels \(\mathrm{S} 20^{\circ} \mathrm{W}\) at \(18 \mathrm{mi} / \mathrm{hr}\). Approximately how far apart are the ships at 3:00 P.M.?

Exer. 11-20: Represent the complex number geometrically. $$ 4(-1+2 i) $$

Exer. 19-22: Find the indicated roots, and represent them geometrically. The eight eighth roots of unity

Exer. 17-20: Determine \(m\) such that the two vectors are orthogonal. $$ 5 m \mathbf{i}+3 \mathbf{j}, \quad 2 \mathbf{i}+7 \mathbf{j} $$

\((m+n) \mathbf{a}=m \mathbf{a}+n \mathbf{a}\)

Flight distance An airplane flies 165 miles from point \(A\) in the direction \(130^{\circ}\) and then travels in the direction \(245^{\circ}\) for 80 miles. Approximately how far is the airplane from \(A\) ?

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

Exer. 19-22: Find the indicated roots, and represent them geometrically. The five fifth roots of \(1+i\)

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. (a) \(\mathbf{a} \cdot(\mathbf{b}+\mathbf{c})\) (b) \(\mathbf{a} \cdot \mathbf{b}+\mathbf{a} \cdot \mathbf{c}\)

Jogger's course A jogger runs at a constant speed of one mile every 8 minutes in the direction \(\mathrm{S} 40^{\circ} \mathrm{E}\) for 20 minutes and then in the direction \(\mathrm{N} 20^{\circ} \mathrm{E}\) for the next 16 minutes. Approximate, to the nearest tenth of a mile, the straightline distance from the endpoint to the starting point of the jogger's course.

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

Exer. 19-22: Find the indicated roots, and represent them geometrically. The five fifth roots of \(-\sqrt{3}-i\)

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. (a) \(\mathbf{b} \cdot(\mathbf{a}-\mathbf{c})\) (b) \(\mathbf{b} \cdot \mathbf{a}-\mathbf{b} \cdot \mathbf{c}\)

Surveying Two points \(P\) and \(Q\) on level ground are on opposite sides of a building. To find the distance between the points, a surveyor chooses a point \(R\) that is 300 feet from \(P\) and 438 feet from \(Q\) and then determines that angle \(P R Q\) has measure \(37^{\circ} 40^{\prime}\) (see the figure). Approximate the distance between \(P\) and \(Q\).

Installing a solar panel Shown in the figure is a solar panel 10 feet in width, which is to be attached to a roof that makes an angle of \(25^{\circ}\) with the horizontal. Approximate the length \(d\) of the brace that is needed for the panel to make an angle of \(45^{\circ}\) with the horizontal.

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

Exer. 23-30: Find the solutions of the equation. $$ x^{4}-16=0 $$

Exer. 21-28: Given that \(a=\langle 2,-3\rangle, \quad b=\langle 3,4\rangle\), and \(c=\langle-1,5\rangle\), find the number. $$ (2 \mathbf{a}+\mathbf{b}) \cdot(3 \mathbf{c}) \quad 24(\mathbf{a}-\mathbf{b}) \cdot(\mathbf{b}+\mathbf{c}) $$