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Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.
5.1583 pounds, 75.8° from the horizontal
For the following exercises, assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Solve each triangle, if possible. Round each answer to the nearest tenth.
$\beta =\mathrm{50\xb0},a=105,b=45$
Not possible
$\alpha =\mathrm{43.1\xb0},a=184.2,b=242.8$
Solve the triangle.
$C=\mathrm{120\xb0},a=23.1,c=34.1$
Find the area of the triangle.
A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, 2.1 km apart, to be 25° and 49°, as shown in [link] . Find the distance of the plane from point $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and the elevation of the plane.
distance of the plane from point $\text{\hspace{0.17em}}A:\text{\hspace{0.17em}}$ 2.2 km, elevation of the plane: 1.6 km
Solve the triangle, rounding to the nearest tenth, assuming $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ s opposite side $c:\text{\hspace{0.17em}}a=4,b=6,c=8.$
Solve the triangle in [link] , rounding to the nearest tenth.
$B=\mathrm{71.0\xb0},C=\mathrm{55.0\xb0},a=12.8$
Find the area of a triangle with sides of length 8.3, 6.6, and 9.1.
To find the distance between two cities, a satellite calculates the distances and angle shown in [link] (not to scale). Find the distance between the cities. Round answers to the nearest tenth.
40.6 km
Plot the point with polar coordinates $\text{\hspace{0.17em}}\left(3,\frac{\pi}{6}\right).$
Plot the point with polar coordinates $\text{\hspace{0.17em}}\left(5,-\frac{2\pi}{3}\right)$
Convert $\text{\hspace{0.17em}}\left(6,-\frac{3\pi}{4}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.
Convert $\text{\hspace{0.17em}}\left(-2,\frac{3\pi}{2}\right)\text{\hspace{0.17em}}$ to rectangular coordinates.
$\text{\hspace{0.17em}}\left(0,2\right)\text{\hspace{0.17em}}$
Convert $\left(7,-2\right)$ to polar coordinates.
Convert $\left(-9,-4\right)$ to polar coordinates.
$\left(9.8489,\mathrm{203.96\xb0}\right)$
For the following exercises, convert the given Cartesian equation to a polar equation.
$x=-2$
${x}^{2}+{y}^{2}=64$
$r=8$
${x}^{2}+{y}^{2}=-2y$
For the following exercises, convert the given polar equation to a Cartesian equation.
$r=7\text{cos}\text{\hspace{0.17em}}\theta $
${x}^{2}+{y}^{2}=7x$
$r=\frac{-2}{4\mathrm{cos}\text{\hspace{0.17em}}\theta +\mathrm{sin}\text{\hspace{0.17em}}\theta}$
For the following exercises, convert to rectangular form and graph.
$\theta =\frac{3\pi}{4}$
$y=-x$
$r=5\mathrm{sec}\text{\hspace{0.17em}}\theta $
For the following exercises, test each equation for symmetry.
$r=4+4\mathrm{sin}\text{\hspace{0.17em}}\theta $
symmetric with respect to the line $\theta =\frac{\pi}{2}$
$r=7$
Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=1-5\mathrm{sin}\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ Label the axis intercepts.
Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=5\mathrm{sin}\left(7\theta \right).$
Sketch a graph of the polar equation $\text{\hspace{0.17em}}r=3-3\mathrm{cos}\text{\hspace{0.17em}}\theta $
For the following exercises, find the absolute value of each complex number.
$-2+6i$
$4-\text{}3i$
5
Write the complex number in polar form.
$5+9i$
$\frac{1}{2}-\frac{\sqrt{3}}{2}\text{}i$
$\mathrm{cis}\left(-\frac{\pi}{3}\right)$
For the following exercises, convert the complex number from polar to rectangular form.
$z=5\mathrm{cis}\left(\frac{5\pi}{6}\right)$
$z=3\mathrm{cis}\left(\mathrm{40\xb0}\right)$
$2.3+1.9i$
For the following exercises, find the product $\text{\hspace{0.17em}}{z}_{1}{z}_{2}\text{\hspace{0.17em}}$ in polar form.
${z}_{1}=2\mathrm{cis}\left(\mathrm{89\xb0}\right)$
${z}_{2}=5\mathrm{cis}\left(\mathrm{23\xb0}\right)$
${z}_{1}=10\mathrm{cis}\left(\frac{\pi}{6}\right)$
${z}_{2}=6\mathrm{cis}\left(\frac{\pi}{3}\right)$
$60\mathrm{cis}\left(\frac{\pi}{2}\right)$
For the following exercises, find the quotient $\text{\hspace{0.17em}}\frac{{z}_{1}}{{z}_{2}}\text{\hspace{0.17em}}$ in polar form.
${z}_{1}=12\mathrm{cis}\left(\mathrm{55\xb0}\right)$
${z}_{2}=3\mathrm{cis}\left(\mathrm{18\xb0}\right)$
${z}_{1}=27\mathrm{cis}\left(\frac{5\pi}{3}\right)$
${z}_{2}=9\mathrm{cis}\left(\frac{\pi}{3}\right)$
$3\mathrm{cis}\left(\frac{4\pi}{3}\right)$
For the following exercises, find the powers of each complex number in polar form.
Find $\text{\hspace{0.17em}}{z}^{4}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=2\mathrm{cis}\left(\mathrm{70\xb0}\right)$
Find $\text{\hspace{0.17em}}{z}^{2}\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}z=5\mathrm{cis}\left(\frac{3\pi}{4}\right)$
$25\mathrm{cis}\left(\frac{3\pi}{2}\right)$
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