Chapter 6

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(-8 - 5\sqrt{3}i\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = \langle 1, 0 \rangle\) \(\mathbf{v} = \langle 0, -2 \rangle\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = \langle 2, 1 \rangle\), \(\mathbf{v} = \langle 1, 3 \rangle\)

In Exercises \(25-34,\) use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$ A=120^{\circ}, \quad a=b=25 $$

In Exercises 15-32, represent the complex number graphically, and find the trigonometric form of the number. \(-9 - 2\sqrt{10}i\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = \langle 3, 2 \rangle\) \(\mathbf{v} = \langle 4, 0 \rangle\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = \langle 2, 3 \rangle\), \(\mathbf{v} = \langle 4, 0 \rangle\)

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. \(A\ =\ 120^{\circ}\), \(a\ =\ 25\), \(b\ =\ 24\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(2(\cos\ 60^{\circ} + i\ \sin\ 60^{\circ})\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 3\mathbf{i} + 4\mathbf{j}\) \(\mathbf{v} = -2\mathbf{j}\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = \langle -5, 3 \rangle\), \(\mathbf{v} = \langle 0, 0 \rangle\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 8\), \(b = 12\), \(c = 17\)

In Exercises \(25-34,\) use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$ A=45^{\circ}, \quad a=b=1 $$

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(5(\cos\ 135^{\circ} + i\ \sin\ 135^{\circ})\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}\) \(\mathbf{v} = \mathbf{i} - 2\mathbf{j}\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = \langle 0, 0 \rangle\), \(\mathbf{v} = \langle 2, 1 \rangle\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 33\), \(b = 36\), \(c = 25\)

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. \(A\ =\ 25^{\circ}4'\), \(a\ =\ 9.5\), \(b\ =\ 22\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(\sqrt{48}[\cos(-30^{\circ}) + i\ \sin(-30^{\circ})]\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 2\mathbf{i} - \mathbf{j}\) \(\mathbf{v} = 6\mathbf{i} + 4\mathbf{j}\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = \mathbf{i} + \mathbf{j}\), \(\mathbf{v} = 2\mathbf{i} - 3\mathbf{j}\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 2.5\), \(b = 10.2\), \(c = 9\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(\sqrt{8}(\cos\ 225^{\circ} + i\ \sin\ 225^{\circ})\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = -6\mathbf{i} - 3\mathbf{j}\) \(\mathbf{v} = -8\mathbf{i} + 4\mathbf{j}\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = -2\mathbf{i} + \mathbf{j}\), \(\mathbf{v} = 3\mathbf{j}\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 75.4\), \(b = 52\), \(c = 52\)

In Exercises \(33-42,\) find the standard form of the complex number. Then represent the complex number graphically. $$ \frac{9}{4}\left(\cos \frac{3 \pi}{4}+i \sin \frac{3 \pi}{4}\right) $$

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 5\mathbf{i} + 5\mathbf{j}\) \(\mathbf{v} = -6\mathbf{i} + 6\mathbf{j}\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = 2\mathbf{i}, \)\mathbf{v} = \mathbf{j}$

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 12.32\), \(b = 8.46\), \(c = 15.05\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(6 \left(\cos\ \dfrac{5\pi}{12} + i\ \sin\ \dfrac{5\pi}{12} \right)\)

In Exercises 31-40, find the angle \(\theta\) between the vectors. \(\mathbf{u} = 2\mathbf{i} - 3\mathbf{j}\) \(\mathbf{v} = 4\mathbf{i} + 3\mathbf{j}\)

In Exercises 31-38, find (a) \(\small{\mathbf{u}} + \small{\mathbf{v}}\), (b) \(\small{\mathbf{u}} - \small{\mathbf{v}}\), and (c) \(\small{2\mathbf{u}} - \small{3\mathbf{v}}\), Then sketch each resultant vector. \(\mathbf{u} = 2\mathbf{j}\), \(\mathbf{v} = 3\mathbf{i}\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 3.05\), \(b = 0.75\), \(c = 2.45\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(7(\cos\ 0 + i\ \sin\ 0)\)

In Exercises \(31-40,\) find the angle \(\theta\) between the vectors. $$ \begin{array}{l}{\mathbf{u}=\cos \left(\frac{\pi}{3}\right) \mathbf{i}+\sin \left(\frac{\pi}{3}\right) \mathbf{j}} \\ {\mathbf{v}=\cos \left(\frac{3 \pi}{4}\right) \mathbf{i}+\sin \left(\frac{3 \pi}{4}\right) \mathbf{j}}\end{array} $$

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{u} = \langle 3, 0 \rangle\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = 1\), \(b = \frac{1}{2}\), \(c = \frac{3}{4}\)

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(C\ =\ 120^{\circ}\), \(a\ =\ 4\), \(b\ =\ 6\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(8 \left(\cos\ \dfrac{\pi}{2} + i\ \sin\ \dfrac{\pi}{2} \right)\)

In Exercises \(31-40,\) find the angle \(\theta\) between the vectors. $$ \begin{array}{l}{\mathbf{u}=\cos \left(\frac{\pi}{4}\right) \mathbf{i}+\sin \left(\frac{\pi}{4}\right) \mathbf{j}} \\ {\mathbf{v}=\cos \left(\frac{\pi}{2}\right) \mathbf{i}+\sin \left(\frac{\pi}{2}\right) \mathbf{j}}\end{array} $$

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. \(\mathbf{u} = \langle 0, -2 \rangle\)

In Exercises 33-40, use Heron's Area Formula to find the area of the triangle. \(a = \frac{3}{5}\), \(b = \frac{5}{8}\), \(c = \frac{3}{8}\)

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(B\ =\ 130^{\circ}\), \(a\ =\ 62\), \(c\ =\ 20\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(5[\cos(198^{\circ}45') + i\ \sin(198^{\circ}45')]\)

In Exercises 41-44, graph the vectors and find the degree measure of the angle \(\theta\) between the vectors. \(\mathbf{u} = 3\mathbf{i} + 4\mathbf{j}\) \(\mathbf{v} = -7\mathbf{i} + 5\mathbf{j}\)

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

NAVIGATION A boat race runs along a triangular course marked by buoys \(A\), \(B\), and \(C\). The race starts with the boats headed west for 3700 meters. The other two sides of the course lie to the north of the first side, and their lengths are 1700 meters and 3000 meters. Draw a figure that gives a visual representation of the situation, and find the bearings for the last two legs of the race.

In Exercises 39-44, find the area of the triangle having the indicated angle and sides. \(A\ =\ 43^{\circ}45'\), \(b\ =\ 57\), \(c\ =\ 85\)

In Exercises 33-42, find the standard form of the complex number. Then represent the complex number graphically. \(9.75[\cos(280^{\circ}30') + i\ \sin(280^{\circ}30')]\)