11.2.2: Homework
- Page ID
- 197627
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reading Questions
- The Law of Cosines is a generalization of what famous theorem?
- Under what two conditions (e.g., SSS, ASA, etc.) is the Law of Cosines useful for solving a triangle when the Law of Sines is not?
- State one of the three versions of the Law of Cosines.
- When using the Law of Cosines to find a missing side, what information must you have?
- When using the Law of Cosines to find a missing angle, what information must you have?
- What is a common mistake regarding the order of operations when simplifying the right side of the Law of Cosines formula?
- After using the Law of Cosines to find one angle in a triangle, what is the recommended method for finding a second angle?
- When using the Law of Sines after using the Law of Cosines, which angle should you choose to find next to avoid the ambiguous case? Why?
Homework
Concept Check
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The Law of Cosines is a generalization of what theorem?
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If you know all three sides of a triangle and one angle, what might be the advantage of using the Law of Cosines to find another angle, instead of the Law of Sines?
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Piyali is solving a triangle in which \(a = 20\), \(b = 16\), and \(A = 26^{\circ}\). She finds that \(\sin \left(B\right) \approx 0.3507\). How does she know that \(B \approx 20.5^{\circ}\), and not \(159.5^{\circ}\)?
Basic Skills
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Simplify \(5^2+7^2-2(5)(7) \cos\left( \theta \right)\).
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Evaluate the expression in part (a) for \(\theta=29^{\circ}\).
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Evaluate the expression in part (a) for \(\theta=151^{\circ}\).
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Simplify \(26.1^2+32.5^2-2(26.1)(32.5) \cos\left( \phi \right)\).
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Evaluate the expression in part (a) for \(\phi=64^{\circ}\).
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Evaluate the expression in part (a) for \(\phi=116^{\circ}\).
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Solve \(b^2=a^2+c^2-2 a c \cos\left( \beta \right)\) for \(\cos\left( \beta \right)\).
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For the equation in part (a), find \(\cos\left( \beta \right)\) if \(a=5\), \(b=11\), and \(c=8\).
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Solve \(a^2=b^2+c^2-2 b c \cos\left( \alpha \right)\) for \(\cos\left( \alpha \right)\).
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For the equation in part (a), find \(\cos\left( \alpha \right)\) if \(a=4.6\), \(b=7.2\), and \(c=9.4\).
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For the following exercises, find the indicated side. Round to two decimal places.
For the following exercises, find the indicated angle. Round to two decimal places.
For the following exercises, find the angles of the triangle. Round answers to two decimal places.
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\(a = 23\), \(b = 14\), \(c = 18\)
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\(a = 18\), \(b = 25\), \(c = 19\)
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\(a = 16.3\), \(b = 28.1\), \(c = 19.4\)
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\(a = 82.3\), \(b = 22.5\), \(c = 66.8\)
For the following exercises, find the unknown side. Round your answers to two decimal places.
For the following exercises, which law should you use to find the labeled unknown value, the Law of Sines or the Law of Cosines? Write an equation you can solve to find the unknown value. For Problems 32 - 35, you may need two steps to find the unknown value.
For the following exercises,
(a) Sketch and label the triangle.
(b) Solve the triangle. Round answers to two decimal places.
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\(B = 47^{\circ}\), \( a = 23\), \(c = 17\)
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\(C = 32^{\circ}\), \(a = 14\), \(b = 18\)
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\(a = 8\), \(b = 7\), \(c = 9\)
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\(a = 23\), \(b = 34\), \(c = 45\)
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\(b = 72\), \(c = 98\), \(B = 38^{\circ}\)
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\(a = 28\), \(c = 41\), \(A = 27^{\circ}\)
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\(c = 5.7\), \(A = 59^{\circ}\), \(B = 82^{\circ}\)
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\(b = 82\), \(A = 11^{\circ}\), \(C = 42^{\circ}\)
Synthesis Questions
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The sides of a parallelogram are 10 inches and 8 inches, and form an angle of \(130^{\circ}\). Find the lengths of the diagonals of the parallelogram.
For the following exercises, find \(x\), the distance from one vertex to the foot of the altitude.
Proofs
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Copy the three figures above showing the three possibilities for an angle \(C\) in a triangle: \(C\) is acute, obtuse, or a right angle. For each figure, explain why it is true that \(c^2=(b-x)^2+y^2\), then rewrite the right side to get \(c^2=\left(x^2+y^2\right)+b^2-2 b x\).
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For each figure, explain why it is true that \(x^2+y^2=a^2\).
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For all three figures, \(a\) is the distance from the origin to the point \((x, y)\). Use the definition of cosine to write \(\cos\left( C \right)\) in terms of \(a\) and \(x\), then solve your equation for \(x\).
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Start with the last equation from (a), and substitute expressions from (b) and (c) to conclude one case of the Law of Cosines.
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Demonstrate the other two cases of the Law of Cosines:\[ \begin{array}{rcl}
a^2 & = & b^2+c^2-2 b c \cos \left(A\right) \\[6pt] & \text{and} & \\[6pt] b^2 & = & a^2+c^2-2 a c \cos \left(B\right) \\[6pt] \end{array} \nonumber \](Hint: See the previous problem and switch the roles of \(a\) and \(c\), etc.) -
Use the Law of Cosines to prove the projection laws:\[ \begin{array}{rcl}
a & = & b \cos\left( C \right)+c \cos\left( B \right) \\[6pt] b & = & c \cos\left( A \right)+a \cos\left( C \right) \\[6pt] c & = & a \cos\left( B \right)+b \cos\left( A \right) \\[6pt] \end{array} \nonumber \]Illustrate with a sketch. (Hint: Add together two of the versions of the Law of Cosines.) -
If \(\triangle A B C\) is isosceles with \(a=b\), show that \(c^2=2 a^2\left(1-\cos\left( C \right)\right)\).
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Use the Law of Cosines to prove:\[ \begin{array}{rcl}
1+\cos\left( A \right) & = &\dfrac{(a+b+c)(-a+b+c)}{2 b c} \\[6pt] 1-\cos\left( A \right) & = & \dfrac{(a-b+c)(-a+b+c)}{2 b c} \\[6pt] \end{array} \nonumber \] -
Prove that\[ \dfrac{\cos\left( A \right)}{a}+\dfrac{\cos\left( B \right)}{b}+\dfrac{\cos\left( C \right)}{c}=\dfrac{a^2+b^2+c^2}{2 a b c}\nonumber \]
Applications
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A surveyor would like to know the distance, \(P Q\), across a small lake, as shown in the figure. She stands at point \(O\) and measures the angle between the lines of sight to points \(P\) and \(Q\) at \(76^{\circ}\). She also finds \(OP = 1400\) meters and \(OQ = 600\) meters. Calculate the distance \(P Q\).
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A geologist wants to measure the diameter of a crater. From her camp, it is 4 miles to the northern-most point of the crater and 2 miles to the southern-most point. If the angle between the two lines of sight is \( 117^{ \circ } \) , what is the diameter of the crater? Round your answer to the nearest hundredth of a mile.
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Highway engineers plan to drill a tunnel through Boney Mountain from \(G\) to \(H\), as shown in the figure. The angle at point \(F\) is \(41^{\circ}\), and the distances to \(G\) and \(H\) are \(900\) yards and \(2500\) yards, respectively. How long will the tunnel be?
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From the Pedimaxus International Airport a tour helicopter can fly to Cliffs of Insanity Point by following a bearing of \(\mathrm{N} \, 8.2^{ \circ } \, \mathrm{E}\) for 192 miles and it can fly to Bigfoot Falls by following a bearing of \(\mathrm{S} \, 68.5^{ \circ } \, \mathrm{E}\) for 207 miles. Find the distance between Cliffs of Insanity Point and Bigfoot Falls. Round your answer to the nearest mile.
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Cliffs of Insanity Point and Bigfoot Falls from Problem 58 above both lie on a straight stretch of the Great Sasquatch Canyon. What bearing would the tour helicopter need to follow to go directly from Bigfoot Falls to Cliffs of Insanity Point? Round your angle to the nearest tenth of a degree.
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Two pilots leave an airport at the same time. One pilot flies at a heading of \(3^{\circ}\) and at a speed of \(320\) miles per hour, the other flies in the heading \(157^{\circ}\) at a speed of \(406\) miles per hour. How far apart are the two pilots after \(3\) hours? What is the heading from the first plane to the second plane at that time?
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Two boats leave port at the same time. One boat sails due west at a speed of \(17\) miles per hour, the other moves in the heading \(42^{\circ}\) at a speed of \(23\) miles per hour. How far apart are the two boats after \(2\) hours? What is the heading from the first boat to the second boat at that time?
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Linda wants to fly directly south from Indianapolis to Cancun, Mexico, a distance of 1290 miles. However, to avoid bad weather, she flies for 400 miles on a heading of \(162^{\circ}\). What is the heading to Cancun from that location, and how far is it?
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Lap sails 8 miles from Key West, Florida on a heading of \(140^{\circ}\). He then changes course and sails for 10 miles due east. What is the heading back to Key West from that point, and how far is it?
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A naturalist sets off on a hike from a lodge on a bearing of \(\mathrm{S} \, 80^{ \circ } \, \mathrm{W}\). After 1.5 miles, she changes her bearing to \(\mathrm{S} \, 17^{ \circ } \, \mathrm{W} \) and continues hiking for 3 miles. Find her distance from the lodge at this point. Round your answer to the nearest hundredth of a mile. What bearing should she follow to return to the lodge? Round your angle to the nearest degree.
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The HMS Sasquatch leaves port on a bearing of \(\mathrm{N} \, 23^{ \circ } \, \mathrm{E}\) and travels for 5 miles. It then changes course and follows a bearing of \(\mathrm{S} \, 41^{ \circ } \, \mathrm{E}\) for 2 miles. How far is it from port? Round your answer to the nearest hundredth of a mile. What is its bearing to port? Round your angle to the nearest degree.
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The SS Bigfoot leaves a harbor bound for Nessie Island which is 300 miles away at a bearing of \(\mathrm{N} \, 32^{ \circ } \, \mathrm{E}\). A storm moves in and after 100 miles, the captain of the Bigfoot finds he has drifted off course. If his bearing to the harbor is now \(\mathrm{S} \, 70^{ \circ } \, \mathrm{W}\), how far is the SS Bigfoot from Nessie Island? Round your answer to the nearest hundredth of a mile. What course should the captain set to head to the island? Round your angle to the nearest tenth of a degree.
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The phone company wants to erect a cell tower on a steep hill inclined \(26^{\circ}\) to the horizontal. The installation crew plans to run a guy wire from a point on the ground 20 feet uphill from the base of the tower and attach it to the tower at a height of 100 feet. How long should the guy wire be?
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Sandstone Peak rises 3500 above the desert. The Park Service plans to run an aerial tramway up the north face, which is inclined at an angle of \(68^{\circ}\) to the horizontal. The base station will be located 500 feet from the foot of Sandstone Peak. Ignoring any slack in the cable, how long should it be?
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From a point 300 feet above level ground in a firetower, a ranger spots two fires in the Yeti National Forest. The angle of depression made by the line of sight from the ranger to the first fire is \( 2.5^{ \circ } \) and the angle of depression made by line of sight from the ranger to the second fire is \( 1.3^{ \circ } \). The angle formed by the two lines of sight is \( 117^{ \circ } \). Find the distance between the two fires. Round your answer to the nearest foot. (Hint: In order to use the \( 117^{ \circ } \) angle between the lines of sight, you will first need to use right angle Trigonometry to find the lengths of the lines of sight. This will allow you to apply the Law of Cosines.)
