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Mathematics LibreTexts

8.R: Further Applications of Trigonometry (Review)

  • Page ID
    18807
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    8.1: Non-right Triangles: Law of Sines

    For the exercises 1-5 assume \(\alpha \) is opposite side \(a\), \(\beta \) is opposite side \(b\), and \(\gamma \) is opposite side \(c\). Solve each triangle, if possible. Round each answer to the nearest tenth.

    1) \(\beta =50^{\circ}, a=105, b=45\)

    Answer

    Not possible

    2) \(\alpha =43.1^{\circ}, a=184.2, b=242.8\)

    3) Solve the triangle.

    Ex 8R 8.1.3.png

    Answer

    \(C=120^{\circ}, a=23.1, c=34.1\)

    4) Find the area of the triangle.

    Ex 8R 8.1.4.png
    Figure 4.

    5) A pilot is flying over a straight Highway. He determines the angles of depression to two mile posts \(2.1\) km apart to be \(25^{\circ}\) and \(49^{\circ}\), as shown in the figure below. Find the distance of the plane from point \(A\) and the elevation of the plane.

    Ex 8R 8.1.5.png

    Answer

    distance of the plane from point \(A:2.2\) km, elevation of the plane: \(1.6\) km

    8.2: Non-right Triangles - Law of Cosines

    1) Solve the triangle, rounding to the nearest tenth, assuming \(\alpha \) is opposite side \(a\), \(\beta \) is opposite side \(b\), and \(\gamma \) s opposite side \(c: a=4, b=6,c=8\).

    2) Solve the triangle in the Figure below, rounding to the nearest tenth.

    Ex 8R 8.2.2.png

    Answer

    \(B=71.0^{\circ},C=55.0^{\circ},a=12.8\)

    3) Find the area of a triangle with sides of length \(8.3\), \(6.6\), and \(9.1\).

    4) To find the distance between two cities, a satellite calculates the distances and angle shown in the Figure below (not to scale). Find the distance between the cities. Round answers to the nearest tenth.

    Ex 8R 8.2.4.png

    Answer

    \(40.6\) km

    8.3: Polar Coordinates

    1) Plot the point with polar coordinates \(\left ( 3,\dfrac{\pi }{6} \right )\).

    2) Plot the point with polar coordinates \(\left ( 5,\dfrac{-2\pi }{3} \right )\).

    Answer

    Ex 8R 8.3.2.png

    3) Convert \(\left ( 6,\dfrac{-3\pi }{4} \right )\) to rectangular coordinates.

    4) Convert \(\left ( -2,\dfrac{3\pi }{2} \right )\) to rectangular coordinates.

    Answer

    \((0,2)\)

    5) Convert \((7,-2)\) to polar coordinates.

    6) Convert \((-9,-4)\) to polar coordinates.

    Answer

    \((9.8489,203.96^{\circ})\)

    For the exercises 7-9, convert the given Cartesian equation to a polar equation.

    7) \(x=-2\)

    8) \(x^2+y^2=64\)

    Answer

    \(r=8\)

    9) \(x^2+y^2=-2y\)

    For the exercises 10-11, convert the given polar equation to a Cartesian equation.

    10) \(r=7\cos \theta\)

    Answer

    \(x^2+y^2=7x\)

    11) \(r=\dfrac{-2}{4\cos \theta +\sin \theta }\)

    For the exercises 12-13, convert to rectangular form and graph.

    12) \(\theta =\dfrac{3\pi }{4}\)

    Answer

    \(y=-x\)

    Ex 8R 8.3.12.png

    13) \(r=5\sec \theta\)

    8.4: Polar Coordinates - Graphs

    For the exercises 1-5, test each equation for symmetry.

    1) \(r=4+4\sin \theta\)

    Answer

    symmetric with respect to the line \(\theta =\dfrac{\pi }{2}\)

    2) \(r=7\)

    3) Sketch a graph of the polar equation \(r=1-5\sin \theta\). Label the axis intercepts.

    Answer

    Ex 8R 8.4.3.png

    4) Sketch a graph of the polar equation \(r=5\sin (7\theta )\).

    5) Sketch a graph of the polar equation \(r=3-3\cos \theta\)

    Answer

    Ex 8R 8.4.5.png

    8.5: Polar Form of Complex Numbers

    For the exercises 1-2, find the absolute value of each complex number.

    1) \(-2+6i\)

    2) \(4-3i\)

    Answer

    \(5\)

    Write the complex number in polar form.

    3) \(5+9i\)

    4) \(\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i\)

    Answer

    \(\mathrm{cis}\left (-\dfrac{\pi }{3} \right )\)

    For the exercises 5-6, convert the complex number from polar to rectangular form.

    5) \(z=5\mathrm{cis}\left (\dfrac{5\pi }{6} \right )\)

    6) \(z=3\mathrm{cis}(40^{\circ})\)

    Answer

    \(2.3+1.9i\)

    For the exercises 7-8, find the product \(z_1 z_2\) in polar form.

    7) \(\begin{align*} z_1 &= 2\mathrm{cis}(89^{\circ})\\ z_2 &= 5\mathrm{cis}(23^{\circ}) \end{align*}\)

    8) \(\begin{align*} z_1 &= 10\mathrm{cis}\left ( \dfrac{\pi }{6} \right )\\ z_2 &= 6\mathrm{cis}\left ( \dfrac{\pi }{3} \right ) \end{align*}\)

    Answer

    \(60\mathrm{cis}\left ( \dfrac{\pi }{2} \right )\)

    For the exercises 9-10, find the quotient \(\dfrac{z_1}{z_2}\) in polar form.

    9) \(\begin{align*} z_1 &= 12\mathrm{cis}(55^{\circ})\\ z_2 &= 3\mathrm{cis}(18^{\circ}) \end{align*}\)

    10) \(\begin{align*} z_1 &= 27\mathrm{cis}\left ( \dfrac{5\pi }{3} \right )\\ z_2 &= 9\mathrm{cis}\left ( \dfrac{\pi }{3} \right ) \end{align*}\)

    Answer

    \(3\mathrm{cis}\left ( \dfrac{4\pi }{3} \right )\)

    For the exercises 11-12, find the powers of each complex number in polar form.

    11) Find \(z^4\) when \(z=2\mathrm{cis}(70^{\circ})\)

    12) Find \(z^2\) when \(z=5\mathrm{cis}\left ( \dfrac{3\pi }{4} \right )\)

    Answer

    \(25\mathrm{cis}\left ( \dfrac{3\pi }{2} \right )\)

    For the exercises 13-14, evaluate each root.

    13) Evaluate the cube root of \(z\) when

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