Chapter 6

In Exercises 41-44, graph the vectors and find the degree measure of the angle \(\theta\) between the vectors. \(\mathbf{u} = 6\mathbf{i} + 3\mathbf{j}\) \(\mathbf{v} = -4\mathbf{i} + 4\mathbf{j}\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{v} = \langle 5, -12 \rangle\)

NAVIGATION A plane flies 810 miles from Franklinto Centerville with a bearing of \(75^{\circ}\). Then it flies 648 miles from Centerville to Rosemount with a bearing of \(32^{\circ}\). Draw a figure that visually represents the situation, and find the straight-line distance and bearing from Franklin to Rosemount.

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(A\ =\ 5^{\circ}15'\), \(b\ =\ 4.5\), \(c\ =\ 22\)

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(5 \left(\cos\ \dfrac{\pi}{9} + i\ \sin\ \dfrac{\pi}{9} \right)\)

In Exercises 41-44, graph the vectors and find the degree measure of the angle \(\theta\) between the vectors. \(\mathbf{u} = 5\mathbf{i} + 5\mathbf{j}\) \(\mathbf{v} = -8\mathbf{i} + 8\mathbf{j}\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{v} = \mathbf{i} + \mathbf{j}\)

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(B\ =\ 72^{\circ}30'\), \(a\ =\ 105\), \(c\ =\ 64\)

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(10 \left(\cos\ \dfrac{2\pi}{5} + i\ \sin\ \dfrac{2\pi}{5} \right)\)

In Exercises 41-44, graph the vectors and find the degree measure of the angle \(\theta\) between the vectors. \(\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}\) \(\mathbf{v} = 8\mathbf{i} + 3\mathbf{j}\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{v} = 6\mathbf{i} - 2\mathbf{j}\)

SURVEYING A triangular parcel of land has 115 meters of frontage, and the other boundaries have lengths of 76 meters and 92 meters. What angles does the frontage make with the two other boundaries?

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(B\ =\ 84^{\circ}30'\), \(a\ =\ 16\), \(b\ =\ 20\)

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(2(\cos\ 155^{\circ} + i\ \sin\ 155^{\circ})\)

In Exercises 45-48, use vectors to find the interior angles of the triangle with the given vertices. \((1, 2)\), \((3, 4)\), \((2, 5)\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{w} = 4\mathbf{j}\)

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

In Exercises 43-46, use a graphing utility to represent the complex number in standard form. \(9(\cos\ 58^{\circ} + i\ \sin\ 58^{\circ})\)

In Exercises 45-48, use vectors to find the interior angles of the triangle with the given vertices. \((-3, -4)\), \((1, 7)\), \((8, 2)\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $\mathbf{w} = -6\mathbf{i}

HEIGHT A flagpole at a right angle to the horizontal is located on a slope that makes an angle of \(12^{\circ}\) with the horizontal. The flagpole's shadow is 16 meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is \(20^{\circ}\). (a) Draw a triangle to represent the situation. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole. (b) Write an equation that can be used to find the height of the flagpole. (c) Find the height of the flagpole.

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\left[2 \left(\cos\ \dfrac{\pi}{4} + i\ \sin\ \dfrac{\pi}{4} \right) \right] \left[6 \left(\cos\ \dfrac{\pi}{12} + i\ \sin\ \dfrac{\pi}{12} \right) \right]\)

In Exercises 45-48, use vectors to find the interior angles of the triangle with the given vertices. \((-1, 0)\), \((2, 2)\), \((0, 6)\)

DISTANCE Two ships leave a port at 9A.M. One travels at a bearing of N \(54^{\circ}\)W at 12 miles per hour, and the other travels at a bearing of S \(67^{\circ}\)W at 16 miles per hour. Approximate how far apart they are at noon that day.

In Exercises \(47-58\) , perform the operation and leave the result in trigonometric form. $$ \left[\frac{3}{4}\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)\right]\left[4\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right)\right] $$

In Exercises 45-48, use vectors to find the interior angles of the triangle with the given vertices. \((-3, 5)\), \((-1, 9)\), \((7, 9)\)

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{w} = 7\mathbf{j} - 3\mathbf{i}\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \([\frac{5}{3}(\cos\ 120^{\circ} + i\ \sin\ 120^{\circ})][\frac{2}{3}(\cos\ 30^{\circ} + i\ \sin\ 30^{\circ})]\)

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 4\), \(||\mathbf{v}|| = 10\), \(\theta = \dfrac{2 \pi}{3}\)

In Exercises 49-52, find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). Magnitude - ||\(\mathbf{v}\)|| \(= 10\) Direction - \(\mathbf{u} = \langle -3, 4 \rangle\)

BRIDGE DESIGN A bridge is to be built across a small lake from a gazebo to a dock (see figure). The bearing from the gazebo to the dock is S \(41^{\circ}\)W. From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are S \(74^{\circ}\)E and S \(28^{\circ}\)E, respectively. Find the distance from the gazebo to the dock.

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \([\frac{1}{2}(\cos\ 100^{\circ} + i\ \sin\ 100^{\circ})][\frac{4}{5}(\cos\ 300^{\circ} + i\ \sin\ 300^{\circ})]\)

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 100\), \(||\mathbf{v}|| = 250\), \(\theta = \dfrac{\pi}{6}\)

In Exercises 49-52, find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). Magnitude - ||\(\mathbf{v}\)|| \(= 3\) Direction - \(\mathbf{u} = \langle -12, -5 \rangle\)

RAILROAD TRACK DESIGN The circular arc of a railroad curve has a chord of length 3000 feet corresponding to a central angle of \(40^{\circ}\). (a) Draw a diagram that visually represents the situation.Show the known quantities on the diagram and use the variables \(r\) and \(s\) to represent the radius of the arc and the length of the arc, respectively. (b) Find the radius \(r\) of the circular arc. (c) Find the length \(s\) of the circular arc.

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \((\cos\ 80^{\circ} + i\ \sin\ 80^{\circ})(\cos\ 330^{\circ} + i\ \sin\ 330^{\circ})\)

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 9\), \(||\mathbf{v}|| = 36\), \(\theta = \dfrac{3 \pi}{4}\)

In Exercises 49-52, find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). Magnitude - ||\(\mathbf{v}\)|| \(= 9\) Direction - \(\mathbf{u} = \langle 2, 5 \rangle\)

GLIDE PATH A pilot has just started on the glide path for landing at an airport with a runway of length 9000 feet. The angles of depression from the plane to the ends of the runway are \(17.5^{\circ}\) and \(18.8^{\circ}\). (a) Draw a diagram that visually represents the situation. (b) Find the air distance the plane must travel until touching down on the near end of the runway. (c) Find the ground distance the plane must travel until touching down. (d) Find the altitude of the plane when the pilot begins the descent.

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \((\cos\ 5^{\circ} + i\ \sin\ 5^{\circ})(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ})\)

In Exercises 49-52, find \(\mathbf{u \cdot v}\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\). \(||\mathbf{u}|| = 4\), \(||\mathbf{v}|| = 12\), \(\theta = \dfrac{\pi}{3}\)

In Exercises 49-52, find the vector \(\mathbf{v}\) with the given magnitude and the same direction as \(\mathbf{u}\). Magnitude - ||\(\mathbf{v}\)|| \(= 8\) Direction - \(\mathbf{u} = \langle 3, 3 \rangle\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{3(\cos\ 50^{\circ} + i\ \sin\ 50^{\circ})}{9(\cos\ 20^{\circ} + i\ \sin\ 20^{\circ})}\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle -12, 30 \rangle\) \(\mathbf{v} = \langle \frac{1}{2}, -\frac{5}{4} \rangle\)

In Exercises 53-56, the initial and terminal points of a vector are given. Write a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Initial Point - \((-2, 1)\) Terminal Point - \((3, -2)\)

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{(\cos\ 120^{\circ} + i\ \sin\ 120^{\circ})}{2(\cos\ 40^{\circ} + i\ \sin\ 40^{\circ})}\)

In Exercises 53-58, determine whether \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, parallel, or neither. \(\mathbf{u} = \langle 3, 15 \rangle\) \(\mathbf{v} = \langle -1, 5 \rangle\)

In Exercises 53-56, the initial and terminal points of a vector are given. Write a linear combination of the standard unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). Initial Point - \((0, -2)\) Terminal Point - \((3, 6)\)

DISTANCE A family is traveling due west on a road that passes a famous landmark. At a given time the bearing to the landmark is N \(62^{\circ}\)W, and after the family travels 5 miles farther the bearing is N \(38^{\circ}\)W. What is the closest the family will come to the landmark while on the road?

In Exercises 47-58, perform the operation and leave the result in trigonometric form. \(\dfrac{\cos\ \pi + i\ \sin\ \pi}{\cos(\pi/3) + i\ \sin(\pi/3)}\)