Chapter 7

Solew the given triangles. The standard notation for labeling of triangles is used. Round answers to four decimal places. $$a=15, c=21, B=100^{\circ}$$

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$C=40^{\circ}, A=80^{\circ}, c=35$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=\langle 4,0\rangle, \mathbf{v}=\langle-1.5,2.5\rangle$$

Express each complex number in trigonometric form. $$2-2 \sqrt{3} i$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 5,-3\rangle, \mathbf{w}=\langle 1,1\rangle$$

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=2 \sin \theta$$

Plot the points, given in polar coordinates, on a polar grid. $$(2,0)$$

Solew the given triangles. The standard notation for labeling of triangles is used. Round answers to four decimal places. $$a=30, b=20, C=87^{\circ}$$

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$C=120^{\circ}, A=25^{\circ}, c=14$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=\left\langle\frac{1}{3}, \frac{2}{5}\right\rangle, \mathbf{v}=\langle 1,2\rangle$$

Express each complex number in trigonometric form. $$4-4 i$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 10,5\rangle, \mathbf{w}=\langle 2,-1\rangle$$

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=-\sin \theta$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(3, \frac{\pi}{4}\right)$$

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$A=130.5^{\circ}, C=20^{\circ}, a=20$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=\left\langle\frac{1}{4}, \frac{1}{2}\right\rangle, \mathbf{v}=\left\langle-\frac{1}{2}, \frac{3}{4}\right\rangle$$

Express each complex number in trigonometric form. $$-5+5 i$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 1,2\rangle, \mathbf{w}=\langle-3,3\rangle$$

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=-2 \cos \theta$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(-2, \frac{\pi}{3}\right)$$

Solew the given triangles. The standard notation for labeling of triangles is used. Round answers to four decimal places. $$c=27, a=18, B=64^{\circ}$$

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$B=63.7^{\circ}, C=48^{\circ}, b=33$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=-2 \mathbf{i}+3 \mathbf{j}, \mathbf{v}=4 \mathbf{i}-\mathbf{j}$$

Express each complex number in trigonometric form. $$2 \sqrt{3}-2 i$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 6,12\rangle, \mathbf{w}=\langle 3,1\rangle$$

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=\frac{3}{2} \cos \theta$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(-1, \frac{\pi}{6}\right)$$

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$C=52.1^{\circ}, A=73^{\circ}, a=15$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=6 \mathbf{i}-2 \mathbf{j}, \mathbf{v}=-5 \mathbf{i}+3 \mathbf{j}$$

Express each complex number in trigonometric form. $$-3 \sqrt{3}+3 i$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle-4,3\rangle, \mathbf{w}=\langle 1,-3\rangle$$

In Exercises \(7-22,\) sketch the graphs of the polar equations. $$r=\sqrt{2} \sin \theta$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$-\left(2, \frac{2 \pi}{3}\right)$$

Solew the given triangles. The standard notation for labeling of triangles is used. Round answers to four decimal places. $$a=6, c=16, B=111^{\circ}$$

Solve the given triangles. The standard notation for labeling of triangles is used. Round all answers to four decimal places. $$A=87.4^{\circ}, B=61^{\circ}, b=19$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=-1.1 \mathbf{i}+4 \mathbf{j}, \mathbf{v}=4 \mathbf{i}+2.4 \mathbf{j}$$

Multiply or divide as indicated, and leave the answer in trigonometric form. $$2\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \cdot 4\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 4,5\rangle, \mathbf{w}=\langle-3,4\rangle$$

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$A=35^{\circ}, a=7, b=5$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(-4, \frac{7 \pi}{6}\right)$$

Find \(\mathbf{u}-\mathbf{v}, \mathbf{u}+2 \mathbf{v},\) and \(-3 \mathbf{u}+\mathbf{v}\). $$\mathbf{u}=2.6 \mathbf{i}+5 \mathbf{j}, \mathbf{v}=-2 \mathbf{i}+3.7 \mathbf{j}$$

Multiply or divide as indicated, and leave the answer in trigonometric form. $$3\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right) \cdot 5\left(\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3}\right)$$

Calculate projev. Then decompose \(\mathbf{v}\) into \(\mathbf{v}_{1}\) and \(\mathbf{v}_{2},\) where \(\mathbf{v}_{1}\) is parallel to \(\mathbf{w}\) and \(\mathbf{v}_{2}\) is orthogonal to \(\mathbf{w}\) $$\mathbf{v}=\langle 6,-3\rangle, \mathbf{w}=\langle 4,2\rangle$$

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$A=25^{\circ}, a=7, b=9$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(\frac{1}{2}, \frac{3 \pi}{2}\right)$$

Find the magnitude and direction of each of the given vectors. Express the direction as an angle \(\theta\) in standard position, where \(0^{\circ} \leq \theta<360^{\circ},\) to tuo decimal places. $$\mathbf{u}=\langle-1,2\rangle$$

Multiply or divide as indicated, and leave the answer in trigonometric form. $$\frac{1}{2}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right) \cdot 3\left(\cos \frac{4 \pi}{3}+i \sin \frac{4 \pi}{3}\right)$$

Determine whether the given pairs of vectors are orthogonal. $$\mathbf{v}=\langle 1,2\rangle, \mathbf{w}=\langle-4,1\rangle$$

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$A=40^{\circ}, a=6, b=5$$

Convert each of the given pairs of polar coordinates to a pair of rectangular coordinates. $$\left(\frac{3}{2}, \frac{\pi}{2}\right)$$