Chapter 7

\(A 10\) -meter telephone pole casts a 17 -meter shadow directly down a slope when the angle of elevation of the sun is \(42^{\circ}\) (see figure). Find \(\theta,\) the angle of elevation of the ground.

Determine whether u and v are orthogonal, parallel, or neither. $$\begin{aligned} &\mathbf{u}=\langle 15,9\rangle\\\ &\mathbf{v}=\langle-5,-3\rangle \end{aligned}$$

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{u}=\langle 0,-2\rangle$$.

Represent the complex number graphically, and find the standard form of the number. $$8\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$$

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=\langle-1,1\rangle$$.

Determine whether u and v are orthogonal, parallel, or neither. $$\begin{aligned} &\mathbf{u}=-\frac{3}{5} \mathbf{i}+\frac{7}{10} \mathbf{j}\\\ &\mathbf{v}=12 \mathbf{i}-14 \mathbf{j} \end{aligned}$$

Represent the complex number graphically, and find the standard form of the number. $$3.75\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)$$

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{v}=\langle-2,2\rangle$$

Determine whether u and v are orthogonal, parallel, or neither. $$\begin{aligned} &\mathbf{u}=-\frac{9}{10} \mathbf{i}+3 \mathbf{j}\\\ &\mathbf{v}=-5 \mathbf{i}+\frac{3}{2} \mathbf{j} \end{aligned}$$

Two ships leave a port at 9 A.M. One travels at a bearing of \(\mathrm{N} 53^{\circ} \mathrm{W}\) at 12 miles per hour, and the other travels at a bearing of \(\mathrm{S} 67^{\circ} \mathrm{W}\) at 16 miles per hour. Approximate how far apart the ships are at noon.

Represent the complex number graphically, and find the standard form of the number. $$1.5\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)$$

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=\langle-24,-7\rangle$$

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=2 \mathbf{i}-k \mathbf{j}\\\ &\mathbf{v}=3 \mathbf{i}+2 \mathbf{j} \end{aligned}$$

A triangular parcel of ground has sides of lengths 725 feet, 650 feet, and 575 feet. Find the measure of the largest angle.

Represent the complex number graphically, and find the standard form of the number. $$2\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)$$

Determine whether the statement is true or false. Justify your answer. If any three sides or angles of an oblique triangle are known, then the triangle can be solved.

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=\langle-9,12\rangle$$.

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=8 \mathbf{i}+4 \mathbf{j}\\\ &\mathbf{v}=2 \mathbf{i}-k \mathbf{j} \end{aligned}$$

Represent the complex number graphically, and find the standard form of the number. $$5\left(\cos 135^{\circ}+i \sin 135^{\circ}\right)$$

Determine whether the statement is true or false. Justify your answer. If a triangle contains an obtuse angle, then it must be oblique.

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1.$$\mathbf{v}=4 \mathbf{i}-3 \mathbf{j}$$.

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{array}{l} \mathbf{u}=\mathbf{i}+4 \mathbf{j} \\ \mathbf{v}=7 k \mathbf{i}-5 \mathbf{j} \end{array}$$

Represent the complex number graphically, and find the standard form of the number. $$\frac{5}{2}\left[\cos \left(-30^{\circ}\right)+i \sin \left(-30^{\circ}\right)\right]$$

Determine whether the statement is true or false. Justify your answer. Two angles and one side of a triangle do not necessarily determine a unique triangle.

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{w}=\mathbf{i}-2 \mathbf{j}$$.

Represent the complex number graphically, and find the standard form of the number. $$\frac{1}{4}\left[\cos \left(-45^{\circ}\right)+i \sin \left(-45^{\circ}\right)\right]$$

Can the Law of sines be used to solve a right triangle? If so, write a short paragraph explaining how to use the Law of sines to solve the following triangle. Is there an easier way to solve the triangle? Explain. \(B=50^{\circ}, \quad C=90^{\circ}, \quad a=10\)

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{w}=2 \mathbf{j}$$.

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=-3 k \mathbf{i}+2 \mathbf{j}\\\ &\mathbf{v}=-6 \mathbf{i} \end{aligned}$$

The lengths of the sides of a triangular garden at a university are approximately 160 feet, 150 feet, and 140 feet. Approximate the area of the garden.

Represent the complex number graphically, and find the standard form of the number. $$\sqrt{12}\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$$

Given \(A=36^{\circ}\) and \(a=5,\) find values of \(b\) such that the triangle has (a) one solution, (b) two solutions, and (c) no solution.

Find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $$\mathbf{w}=-3 \mathbf{i}$$.

Find the value of \(k\) such that the vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. $$\begin{aligned} &\mathbf{u}=4 \mathbf{i}-4 k \mathbf{j}\\\ &\mathbf{v}=3 \mathbf{j} \end{aligned}$$

Represent the complex number graphically, and find the standard form of the number. $$\sqrt{48}(\cos 0+i \sin 0)$$

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle 2,2\rangle\\\ &\mathbf{v}=\langle 6,1\rangle \end{aligned}$$

Represent the complex number graphically, and find the standard form of the number. $$3\left[\cos \left(18^{\circ} 45^{\prime}\right)+i \sin \left(18^{\circ} 45^{\prime}\right)\right]$$

Use the given values to find the values of the remaining four trigonometric functions of \(\theta\) \(\cos \theta=\frac{5}{13}, \quad \sin \theta=-\frac{12}{13}\)

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle 4,2\rangle\\\ &\mathbf{v}=\langle 1,-2\rangle \end{aligned}$$

Represent the complex number graphically, and find the standard form of the number. $$6\left[\cos \left(230^{\circ} 30^{\prime}\right)+i \sin \left(230^{\circ} 30^{\prime}\right)\right]$$

Use the given values to find the values of the remaining four trigonometric functions of \(\theta\) \(\tan \theta=-\frac{8}{15}, \quad \csc \theta=\frac{17}{8}\)

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle 0,3\rangle\\\ &\mathbf{v}=\langle 2,15\rangle \end{aligned}$$

Determine whether the statement is true or false. Justify your answer. Two sides and their included angle determine a unique triangle.

Use a graphing utility to represent the complex number in standard form. $$5\left(\cos \frac{7 \pi}{9}+i \sin \frac{7 \pi}{9}\right)$$

Write the product as a sum or difference. \(6 \sin 8 \theta \cos 3 \theta\)

Find the projection of \(\mathbf{u}\) onto \(\mathbf{v}\). Then write \(\mathbf{u}\) as the sum of two orthogonal vectors, one of which is \({\mathrm{proj}_v}\mathrm{u}.\) $$\begin{aligned} &\mathbf{u}=\langle-5,-1\rangle\\\ &\mathbf{v}=\langle-1,1\rangle \end{aligned}$$

Determine whether the statement is true or false. Justify your answer. In Heron's Area Formula, \(s\) is the average of the lengths of the three sides of the triangle.

Use a graphing utility to represent the complex number in standard form. $$12\left(\cos \frac{3 \pi}{5}+i \sin \frac{3 \pi}{5}\right)$$

Write the product as a sum or difference. \(2 \cos 2 \theta \cos 5 \theta\)

Use the Law of cosines to prove each identity. (a) \(\frac{1}{2} b c(1+\cos A)=\left(\frac{a+b+c}{2}\right)\left(\frac{-a+b+c}{2}\right)\). (b) \(\frac{1}{2} b c(1-\cos A)=\left(\frac{a-b+c}{2}\right)\left(\frac{a+b-c}{2}\right)\).