Pre-Calc/Trig Unit 3: Analytic Trig and Applications/7.1 and 7.2 Law of Sines and Area of Triangle Area of painted triangle A painter is going to apply a special coating to a triangular metal plate on a new building. Two sides measure are 13.2 m and 5.4 m. She knows that the angle between these sides is «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»#«/mo»«mi»a«/mi»«mo»§#160;«/mo»«mo»§#176;«/mo»«/math». What is the area of the surface she plans to cover with the coating? Round to the nearest square meter.

]]> Did you use «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»A«/mi»«mo»=«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«mi»b«/mi»«mi»c«/mi»«mo»§#160;«/mo»«mi»S«/mi»«mi»i«/mi»«mi»n«/mi»«mo»§#160;«/mo»«mi»A«/mi»«/math» ? The angle must be between the two known sides.

]]> 1.0000000 0.3333333 0 0 #answer]]> <question><wirisCasSession><![CDATA[<session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="en">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>24</mn><mo>,</mo><mn>32</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>answer</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>*</mo><mn>13</mn><mo>.</mo><mn>2</mn><mo>*</mo><mn>5</mn><mo>.</mo><mn>4</mn><mo>*</mo><mi>sin</mi><mo>(</mo><mi>a</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>a</mi><mi>ns</mi><mi>w</mi><mi mathvariant="normal">e</mi><mi>r</mi></math>]]></correctAnswer><correctAnswer id="1"></correctAnswer><correctAnswer id="2"></correctAnswer><correctAnswer id="3"></correctAnswer></correctAnswers><assertions><assertion name="syntax_quantity"><param name="units"><![CDATA[m, s, g, sr, E, K, mol, cd, rad, h, min, l, N, Pa, Hz, W, J, C, V, Ω, F, S, Wb, b, H, T, lx, lm, Gy, Bq, Sv, kat]]></param><param name="unitprefixes">M, k, c, m, y, z, a, f, p, n, µ, d, da, h, G, T, P, E, Z, Y</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-0)</option><option name="relative_tolerance">false</option><option name="precision">2</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession">null</data></localData></question> Area of triangle Find the area of triangle ABC with the given parts. Round to the nearest tenth.

A = 26.4°
b = #a in.
c = #b in.

]]> 1.0000000 0.3333333 0 0 #answer]]> <question><wirisCasSession><![CDATA[<session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="en">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>10</mn><mo>.</mo><mn>0</mn><mo>,</mo><mn>13</mn><mo>.</mo><mn>1</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>12</mn><mo>.</mo><mn>1</mn><mo>,</mo><mn>14</mn><mo>.</mo><mn>1</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>answer</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi><mo>*</mo><mi>sin</mi><mo>(</mo><mn>26</mn><mo>.</mo><mn>4</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>a</mi><mi>ns</mi><mi>w</mi><mi mathvariant="normal">e</mi><mi>r</mi></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_quantity"><param name="units"><![CDATA[m, s, g, sr, E, K, mol, cd, rad, h, min, l, N, Pa, Hz, W, J, C, V, Ω, F, S, Wb, b, H, T, lx, lm, Gy, Bq, Sv, kat]]></param><param name="unitprefixes">M, k, c, m, y, z, a, f, p, n, µ, d, da, h, G, T, P, E, Z, Y</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-1)</option><option name="relative_tolerance">false</option><option name="precision">3</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession">null</data></localData></question> Distance of ship from port It is #a km from Lighthouse A to Port B. The bearing of the port from the lighthouse is N «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»#«/mo»«mi»b«/mi»«mo»§#160;«/mo»«mo»§#176;«/mo»«/math»E. A ship has sailed due west from the port and its bearing from the lighthouse is N31°E. How far has the ship sailed from the port? Round to the nearest tenth of a kilometer.

]]> Draw a diagram and use the Law of Sines.

]]> 1.0000000 0.3333333 0 0 #answer]]> <question><wirisCasSession><![CDATA[<session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="en">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>3</mn><mo>.</mo><mn>9</mn><mo>,</mo><mn>5</mn><mo>.</mo><mn>2</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>70</mn><mo>,</mo><mn>75</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>answer</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>sin</mi><mo>(</mo><mi>b</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>-</mo><mn>31</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo><mo>)</mo></mrow><mrow><mo>(</mo><mi>sin</mi><mo>(</mo><mn>121</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo><mo>)</mo></mrow></mfrac></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>a</mi><mi>ns</mi><mi>w</mi><mi mathvariant="normal">e</mi><mi>r</mi></math>]]></correctAnswer><correctAnswer id="1"></correctAnswer></correctAnswers><assertions><assertion name="syntax_quantity"><param name="units"><![CDATA[m, s, g, sr, E, K, mol, cd, rad, h, min, l, N, Pa, Hz, W, J, C, V, Ω, F, S, Wb, b, H, T, lx, lm, Gy, Bq, Sv, kat]]></param><param name="unitprefixes">M, k, c, m, y, z, a, f, p, n, µ, d, da, h, G, T, P, E, Z, Y</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-1)</option><option name="relative_tolerance">false</option><option name="precision">2</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession">null</data></localData></question> Find the missing parts of the triangle. Find the missing parts of the triangle.

]]> 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1.0000000 0.3333333 0 true true abc no such triangle Incorrect ]]> 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Incorrect ]]> 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 Correct B = 60°, C = 90°, b = 9 Incorrect Find the missing parts of the triangle.B = 11.4°b = 10.38a = 26.26 Find the missing parts of the triangle.

B = 11.4°
b = 10.38
a = 26.26]]> 1.0000000 0.3333333 0 true true abc 1 = 30°, C₁ = 138.6°, c₁ = 34.73;
A₂ = 150°, C₂ = 18.6°, c₂ = 16.75]]> Correct no such triangle Incorrect A = 150°, C = 18.6°, c = 16.75 Incorrect A = 30°, C = 138.6°, c = 34.73 Incorrect Find the missing parts of the triangle.B = 42°30'b = 8.67a = 21.01 Find the missing parts of the triangle.

B = 42°30'
b = 8.67
a = 21.01]]> 1.0000000 0.3333333 0 true true abc A = 40°15', C = 97°15', c = 29.68 Incorrect A = 38°15', C = 99°15', c = 26.68 Incorrect A = 41°15', C = 98°15', c = 31.18 Incorrect no such triangle Correct Height of airplane An airplane is sighted at the same time by two ground observers who are #b miles apart and both to the west side of the airplane. They report the angles of elevation as 14° and «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mo»#«/mo»«mi»a«/mi»«mo»§#160;«/mo»«mo»§#176;«/mo»«/math» . How high is the airplane?

]]> Draw a diagram. Find one of the missing sides with Law of Sines. Then that side becomes the hypotenuse for a right triangle showing the height of the airplane.

]]> 1.0000000 0.3333333 0 0 #answer]]> <question><wirisCasSession><![CDATA[<session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="en">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>18</mn><mo>,</mo><mn>23</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mn>5</mn><mo>*</mo><mo>(</mo><mi>sin</mi><mo>(</mo><mn>14</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo><mo>)</mo></mrow><mrow><mo>(</mo><mi>sin</mi><mo>(</mo><mi>a</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>-</mo><mn>14</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo><mo>)</mo></mrow></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>answer</mi><mo>=</mo><mi>r</mi><mo>*</mo><mo>(</mo><mi>sin</mi><mo>(</mo><mi>a</mi><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo><mo>)</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>a</mi><mi>ns</mi><mi>w</mi><mi mathvariant="normal">e</mi><mi>r</mi></math>]]></correctAnswer><correctAnswer id="1"></correctAnswer></correctAnswers><assertions><assertion name="syntax_quantity"><param name="units"><![CDATA[m, s, g, sr, E, K, mol, cd, rad, h, min, l, N, Pa, Hz, W, J, C, V, Ω, F, S, Wb, b, H, T, lx, lm, Gy, Bq, Sv, kat]]></param><param name="unitprefixes">M, k, c, m, y, z, a, f, p, n, µ, d, da, h, G, T, P, E, Z, Y</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-1)</option><option name="relative_tolerance">false</option><option name="precision">2</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession">null</data></localData></question> Length of guy wire A guy wire to a tower makes a 66° angle with level ground. At a point #a ft farther from the tower than the wire but on the same side of the base as the wire, the angle of elevation to the top of the pole is 36°. Find the wire length (to the nearest foot).

]]> Draw a diagram. Use the Law of Sines to find one of the missing sides of the triangle. Then use that side as the hypotenuse of a right triangle to find the height of the tower, which is the opposite side of this right triangle.

]]> 1.0000000 0.3333333 0 0 #answer]]> <question><wirisCasSession><![CDATA[<session lang="en" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="en">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>random</mi><mo>(</mo><mn>30</mn><mo>,</mo><mn>40</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>answer</mi><mo>=</mo><mfrac><mrow><mi>a</mi><mo>*</mo><mi>sin</mi><mo>(</mo><mn>36</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo></mrow><mrow><mi>sin</mi><mo>(</mo><mn>30</mn><csymbol definitionURL="http://.../units/degree/angular">°</csymbol><mo>)</mo></mrow></mfrac><mo>*</mo><mn>1</mn><mo>.</mo></math></input></command></group></library></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>a</mi><mi>ns</mi><mi>w</mi><mi mathvariant="normal">e</mi><mi>r</mi></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_quantity"><param name="units"><![CDATA[m, s, g, sr, E, K, mol, cd, rad, h, min, l, N, Pa, Hz, W, J, C, V, Ω, F, S, Wb, b, H, T, lx, lm, Gy, Bq, Sv, kat]]></param><param name="unitprefixes">M, k, c, m, y, z, a, f, p, n, µ, d, da, h, G, T, P, E, Z, Y</param></assertion><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-0)</option><option name="relative_tolerance">false</option><option name="precision">2</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession">null</data></localData></question> Solve the triangle.B = 34.4°C = 114.2°b = 29.50 Solve the triangle.

B = 34.4°
C = 114.2°
b = 29.50]]> 1.0000000 0.3333333 0 true true abc A = 29.4°, a = 47.63, c = 27.20 Incorrect A = 31.4°, a = 29.20, c = 49.63 Incorrect A = 29.4°, a = 49.63, c = 29.20 Incorrect A = 31.4°, a = 27.20, c = 47.63 Correct Tracking Station Two tracking stations are on the equator #m miles apart. A weather balloon is located on a bearing of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»N«/mi»«mo»§#160;«/mo»«msup»«mn»42«/mn»«mi»o«/mi»«/msup»«mo»§#160;«/mo»«mi»E«/mi»«/math» from the western station and on a bearing of «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»N«/mi»«mo»§#160;«/mo»«msup»«mn»22«/mn»«mi»o«/mi»«/msup»«mi»E«/mi»«/math» from the eastern station. How far is the balloon from the western station? (Round answer to nearest mile)

]]> Draw a diagram. Find the angle between the eastern station and the western station by adding 90 to 22. Then use the Law of Sines.