Chapter 7

Use a graphing utility to represent the complex number in standard form. $$9\left(\cos 58^{\circ}+i \sin 58^{\circ}\right)$$

Write the product as a sum or difference. \(\frac{1}{3} \cos \frac{\pi}{6} \sin \frac{5 \pi}{3}\)

Use a graphing utility to represent the complex number in standard form. $$2\left(\cos 73^{\circ}+i \sin 73^{\circ}\right)$$

Write the product as a sum or difference. \(\frac{5}{2} \sin \frac{3 \pi}{4} \sin \frac{5 \pi}{6}\)

Describe how the Law of cosines can be used to solve the ambiguous case of the oblique triangle \(A B C,\) where \(a=12\) feet, \(b=30\) feet, and \(A=20^{\circ} .\) Is the result the same as when the Law of sines is used to solve the triangle? Describe the advantages and the disadvantages of each method.

Perform the operation and leave the result in trigonometric form. $$\left[3\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right)\right]\left[9\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)\right]$$

Use a half-angle formula and the Law of cosines to show that, for any triangle, (a) \(\cos \frac{C}{2}=\sqrt{\frac{s(s-c)}{a b}}\) and (b) \(\sin \frac{C}{2}=\sqrt{\frac{(s-a)(s-b)}{a b}}\) where \(s=\frac{1}{2}(a+b+c)\).

Perform the operation and leave the result in trigonometric form. $$\left[4\left(\cos \frac{\pi}{12}+i \sin \frac{\pi}{12}\right)\right]\left[5\left(\cos \frac{7 \pi}{12}+i \sin \frac{7 \pi}{12}\right)\right]$$

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}.\) (There are many correct answers.) $$\mathbf{u}=\langle 2,6\rangle$$

Evaluate the expression without using a calculator. $$\arcsin (-1)$$

Perform the operation and leave the result in trigonometric form. $$\left[\frac{2}{3}\left(\cos \frac{6 \pi}{7}+i \sin \frac{6 \pi}{7}\right)\right]\left[9\left(\cos \frac{9 \pi}{14}+i \sin \frac{9 \pi}{14}\right)\right]$$

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}.\) (There are many correct answers.) $$\mathbf{u}=\langle-7,5\rangle$$

Evaluate the expression without using a calculator. $$\arccos \left(-\frac{\sqrt{3}}{2}\right)$$

Perform the operation and leave the result in trigonometric form. $$\left[\frac{3}{2}\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)\right]\left[6\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)\right]$$

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}.\) (There are many correct answers.) $$\mathbf{u}=\frac{1}{2} \mathbf{i}-\frac{3}{4} \mathbf{j}$$

Evaluate the expression without using a calculator. $$\tan ^{-1} \frac{\sqrt{3}}{3}$$

Perform the operation and leave the result in trigonometric form. $$\left[\frac{5}{3}\left(\cos 140^{\circ}+i \sin 140^{\circ}\right)\right]\left[\frac{2}{3}\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\right]$$

Find two vectors in opposite directions that are orthogonal to the vector \(\mathbf{u}.\) (There are many correct answers.) $$\mathbf{u}=-\frac{5}{2} \mathbf{i}-3 \mathbf{j}$$

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\).$$\mathbf{v}=\frac{3}{4} \mathbf{w}$$

Evaluate the expression without using a calculator. $$\tan ^{-1} \sqrt{3}$$

Perform the operation and leave the result in trigonometric form. $$\left[\frac{1}{2}\left(\cos 115^{\circ}+i \sin 115^{\circ}\right)\right]\left[\frac{4}{5}\left(\cos 300^{\circ}+i \sin 300^{\circ}\right)\right]$$

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\).$$\mathbf{v}=\mathbf{u}+2 \mathbf{w}$$

Perform the operation and leave the result in trigonometric form. $$\left(\cos 290^{\circ}+i \sin 290^{\circ}\right)\left(\cos 200^{\circ}+i \sin 200^{\circ}\right)$$

Find the work done in moving a particle from \(P\) to \(Q\) when the magnitude and direction of the force are given by \(\mathbf{v}.\) $$P=(1,3), \quad Q=(-3,5), \quad \mathbf{v}=-2 \mathbf{i}+3 \mathbf{j}$$

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=-\mathbf{u}+\mathbf{w}$$

Perform the operation and leave the result in trigonometric form. $$\left(\cos 5^{\circ}+i \sin 5^{\circ}\right)\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)$$

The vector \(\mathbf{u}=\langle 1225,2445\rangle\) gives the numbers of hours worked by employees of a temp agency at two pay levels. The vector \(\mathbf{v}=\langle 12.00,10.25\rangle\) gives the hourly wage (in dollars) paid at each level, respectively. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and explain its meaning in the context of the problem. (b) Identify the vector operation used to increase wages by 2 percent.

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=\frac{1}{2}(3 \mathbf{u}+\mathbf{w})$$

Perform the operation and leave the result in trigonometric form. $$\frac{\cos 50^{\circ}+i \sin 50^{\circ}}{\cos 20^{\circ}+i \sin 20^{\circ}}$$

The vector \(\mathbf{u}=\langle 3240,2450\rangle\) gives the numbers of hamburgers and hot dogs, respectively, sold at a fast food stand in one week. The vector \(\mathbf{v}=\langle 3.25,3.50\rangle\) gives the prices in dollars of the food items. (a) Find the dot product \(\mathbf{u} \cdot \mathbf{v}\) and explain its meaning in the context of the problem. (b) Identify the vector operation used to increase prices by \(2 \frac{1}{2}\) percent.

Find the component form of \(v\) and sketch the specified vector operations geometrically, where \(\mathbf{u}=2 \mathbf{i}-\mathbf{j}\) and \(\mathbf{w}=\mathbf{i}+2 \mathbf{j}\). $$\mathbf{v}=2(\mathbf{u}-\mathbf{w})$$

Perform the operation and leave the result in trigonometric form. $$\frac{2\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)}{4\left(\cos 40^{\circ}+i \sin 40^{\circ}\right)}$$

A truck with a gross weight of 30,000 pounds is parked on a slope of \(d^{\circ}\) (see figure). Assume that the only force to overcome is the force of gravity. (a) Find the force required to keep the truck from rolling down the hill in terms of the slope \(d .\) (b) Use a graphing utility to complete the table. $$\begin{array}{|l|l|l|l|l|l|l|} \hline d & 0^{\circ} & 1^{\circ} & 2^{\circ} & 3^{\circ} & 4^{\circ} & 5^{\circ} \\ \hline \text { Force } & & & & & & \\ \hline \end{array}$$ $$\begin{array}{|l|l|l|l|l|l|} \hline d & 6^{\circ} & 7^{\circ} & 8^{\circ} & 9^{\circ} & 10^{\circ} \\ \hline \text { Force } & & & & & \\ \hline \end{array}$$ (c) Find the force perpendicular to the hill when \(d=5^{\circ}.\)

Find the magnitude and direction angle of the vector v. $$\mathbf{v}=5\left(\cos 60^{\circ} \mathbf{i}+\sin 60^{\circ} \mathbf{j}\right)$$

Perform the operation and leave the result in trigonometric form. $$\frac{5(\cos 2 \pi+i \sin 2 \pi)}{4(\cos \pi+i \sin \pi)}$$

A sport utility vehicle with a gross weight of 5400 pounds is parked on a slope of \(10^{\circ} .\) Assume that the only force to overcome is the force of gravity. Find the force required to keep the vehicle from rolling down the hill. Find the force perpendicular to the hill.

Find the magnitude and direction angle of the vector v. $$\mathbf{v}=8\left(\cos 135^{\circ} \mathbf{i}+\sin 135^{\circ} \mathbf{j}\right)$$

Perform the operation and leave the result in trigonometric form. $$\frac{\cos \left(\frac{7 \pi}{4}\right)+i \sin \left(\frac{7 \pi}{4}\right)}{\cos \pi+i \sin \pi}$$

Find the magnitude and direction angle of the vector v. $$\mathbf{v}=12 \mathbf{i}+15 \mathbf{j}$$

Perform the operation and leave the result in trigonometric form. $$\frac{18\left(\cos 54^{\circ}+i \sin 54^{\circ}\right)}{3\left(\cos 102^{\circ}+i \sin 102^{\circ}\right)}$$

A ski patroller pulls a rescue toboggan across a flat snow surface by exerting a constant force of 35 pounds on a handle that makes a constant angle of \(22^{\circ}\) with the horizontal (see figure). Find the work done in pulling the toboggan 200 feet.

Find the magnitude and direction angle of the vector v.$$\mathbf{v}=8 \mathbf{i}-3 \mathbf{j}$$

Perform the operation and leave the result in trigonometric form. $$\frac{9\left(\cos 20^{\circ}+i \sin 20^{\circ}\right)}{5\left(\cos 75^{\circ}+i \sin 75^{\circ}\right)}$$

A force of 50 pounds, exerted at an angle of \(25^{\circ}\) with the horizontal, is required to slide a desk across a floor. Determine the work done in sliding the desk 15 feet.

Find the magnitude and direction angle of the vector v.$$\mathbf{v}=-2 \mathbf{i}+5 \mathbf{j}$$

A mover exerts a horizontal force of 25 pounds on a crate as it is pushed up a ramp that is 12 feet long and inclined at an angle of \(20^{\circ}\) above the horizontal. Find the work done in pushing the crate up the ramp.

Find the magnitude and direction angle of the vector v.$$\mathbf{v}=-7 \mathbf{i}-6 \mathbf{j}$$

Find the component form of v given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch v. Angle:\begin{aligned}&\theta=0^{\circ}\\\&\theta=45^{\circ}\\\&\theta=120^{\circ}\\\ &\theta=135^{\circ}\\\&\theta=150^{\circ}\\\&\theta=90^{\circ}\\\&\mathbf{v} \text { in the direction } \mathbf{i}+3 \mathbf{j}\\\&\mathbf{v} \text { in the direction } 3 \mathbf{i}+4 \mathbf{j} \end{aligned}. Magnitude:$$\|\mathbf{v}\|=3$$

Determine whether the statement is true or false. Justify your answer. The vectors \(\mathbf{u}=\langle 0,0\rangle\) and \(\mathbf{v}=\langle-12,6\rangle\) are orthogonal.

Find the component form of v given its magnitude and the angle it makes with the positive \(x\) -axis. Sketch v. Angle:$$\begin{aligned} &\theta=0^{\circ}\\\ &\theta=45^{\circ}\\\ &\theta=120^{\circ}\\\ &\theta=135^{\circ}\\\ &\theta=150^{\circ}\\\ &\theta=90^{\circ}\\\ &\mathbf{v} \text { in the direction } \mathbf{i}+3 \mathbf{j}\\\ &\mathbf{v} \text { in the direction } 3 \mathbf{i}+4 \mathbf{j} \end{aligned}$$ Magnitude:$$\|\mathbf{v}\|=1$$