Section 1
Law of Sines
Fill in the blank(s).A(n) _______ triangle has no right angles.
Fill in the blank(s).Law of sines: $\frac{a}{\sin A}=$ ______________$=\frac{c}{\sin C}$
Fill in the blank(s).To find the area of any triangle, use one of the three formulas: Area $=$ _____________, ___________or ___________.
Fill in the blank(s).Two___________ and one_____________determine a unique triangle.
Which two cases can be solved using the Law of Sines?
Is the longest side of an oblique triangle always opposite the largest angle of the triangle?
Use the Law of sines to solve the triangle.
Use the Law of sines to solve the triangle.$A=36^{\circ}, \quad a=8, \quad b=5$
Use the Law of sines to solve the triangle.$A=76^{\circ}, \quad a=34, \quad b=21$
Use the Law of sines to solve the triangle.$A=35^{\circ}, \quad B=40^{\circ}, \quad c=10$
Use the Law of sines to solve the triangle.$A=120^{\circ}, \quad B=45^{\circ}, \quad c=16$
Use the Law of sines to solve the triangle.$A=110^{\circ}, \quad a=125, \quad b=100$
Use the Law of sines to solve the triangle.$A=145^{\circ}, \quad a=14, \quad b=4$
Use the Law of sines to solve the triangle.$A=102.4^{\circ}, \quad C=16.7^{\circ}, \quad a=21.6$
Use the Law of sines to solve the triangle.$A=24.3^{\circ}, \quad C=54.6^{\circ}, \quad c=2.68$
Use the Law of sines to solve the triangle.$B=28^{\circ}, \quad C=104^{\circ}, \quad a=3 \frac{5}{8}$
Use the Law of sines to solve the triangle.$A=55^{\circ}, \quad B=42^{\circ}, \quad c=\frac{3}{4}$
Use the Law of sines to solve the triangle.$A=110^{\circ} 15^{\prime}, \quad a=48, \quad b=16$
Use the Law of sines to solve the triangle.$B=2^{\circ} 45^{\prime}, \quad b=6.2, \quad c=5.8$
Use the Law of Sines to solve the triangle. If two solutions exist, find both.$A=76^{\circ}, \quad a=18, \quad b=20$
Use the Law of Sines to solve the triangle. If two solutions exist, find both.$A=110^{\circ}, \quad a=125, \quad b=200$
Use the Law of Sines to solve the triangle. If two solutions exist, find both.$A=120^{\circ}, \quad a=b=25$
Use the Law of Sines to solve the triangle. If two solutions exist, find both.$A=60^{\circ}, \quad a=9, \quad c=10$
Use the Law of Sines to solve the triangle. If two solutions exist, find both.$A=58^{\circ}, \quad a=11.4, \quad b=12.8$
Use the Law of Sines to solve the triangle. If two solutions exist, find both.$A=58^{\circ}, \quad a=4.5, \quad b=12.8$
Find the value(s) of $b$ such that the triangle has (a) one solution,(b) two solutions, and (c) no solution.$A=36^{\circ}, \quad a=5$
Find the value(s) of $b$ such that the triangle has (a) one solution,(b) two solutions, and (c) no solution.$A=60^{\circ}, \quad a=10$
Find the value(s) of $b$ such that the triangle has (a) one solution,(b) two solutions, and (c) no solution.$A=10^{\circ}, \quad a=10.8$
Find the value(s) of $b$ such that the triangle has (a) one solution,(b) two solutions, and (c) no solution.$A=88^{\circ}, \quad a=315.6$
Find the area of the triangle having the indicated angle and sides.$C=110^{\circ}, \quad a=6, \quad b=10$
Find the area of the triangle having the indicated angle and sides.$B=130^{\circ}, \quad a=92, \quad c=30$
Find the area of the triangle having the indicated angle and sides.$A=150^{\circ}, \quad b=8, \quad c=10$
Find the area of the triangle having the indicated angle and sides.$C=120^{\circ}, \quad a=4, \quad b=6$
Find the area of the triangle having the indicated angle and sides.$B=75^{\circ} 15^{\prime}, \quad a=103, \quad c=58$
Find the area of the triangle having the indicated angle and sides.$C=85^{\circ} 45^{\prime}, \quad a=16, \quad b=20$
Because of prevailing winds, a tree grew so that it is leaning $4^{\circ}$ from the vertical. At a point40 meters from the tree, the angle of elevation to the top of the tree is $30^{\circ}$ (see figure). Find the height $h$ of the tree.
A bridge is to be built across a small lake from a gazebo to a dock. The bearing from the gazebo to the dock is $\mathrm{S} 41^{\circ} \mathrm{W}$. From a tree 100 meters from the gazebo, the bearings to the gazebo and the dock are $\mathrm{S} 74^{\circ} \mathrm{E}$ and $\mathrm{S} 28^{\circ} \mathrm{E},$ respectively (see figure). Find the distance from the gazebo to the dock.
A plane flies 500 kilometers with a bearing of $316^{\circ}$ (clockwise from north) from Naples to Elgin. The plane then flies 720 kilometers from Elgin to Canton (see figure). Canton is due west of Naples. Find the bearing of the flight from Elgin to Canton.
A flagpole at a right angle to the horizontal is located on a slope that makes an angle of $12^{\circ}$ with the horizontal. The flagpole casts a 16 -meter shadow up the slope when the angle of elevation from the tip of the shadow to the sun is $20^{\circ} .$(a) Draw a triangle that represents the problem. Show the known quantities on the triangle and use a variable to indicate the height of the flagpole.(b) Write an equation involving the unknown quantity.(c) Find the height of the flagpole.
The angles of elevation $\theta$ and $\phi$ to an airplane are being continuously monitored at two observation points $A$ and $B,$ respectively, which are 2 miles apart, and the airplane is east of both points in the same vertical plane.(a) Draw a diagram that illustrates the problem.(b) Write an equation giving the distance $d$ between the plane and point $B$ in terms of $\theta$ and $\phi$(c) Use the equation from part (b) to find the distance between the plane and point $B$ when $\theta=40^{\circ}$ and $\phi=60^{\circ}$
The bearing from the Pine Knob fire tower to the Colt Station fire tower is $\mathrm{N} 65^{\circ} \mathrm{E},$ and the two towers are 30 kilometers apart. $\mathrm{A}$ fire spotted by rangers in each tower has a bearing of $\mathrm{N} 80^{\circ} \mathrm{E}$ from Pine Knob and $\mathrm{S} 70^{\circ} \mathrm{E}$ from Colt Station (see figure). Find the distance of the fire from each tower.
$A 10$ -meter telephone pole casts a 17 -meter shadow directly down a slope when the angle of elevation of the sun is $42^{\circ}$ (see figure). Find $\theta,$ the angle of elevation of the ground.
A boat is sailing due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to a lighthouse is $\mathrm{S} 70^{\circ} \mathrm{E},$ and15 minutes later, the bearing is $\mathrm{S} 63^{\circ} \mathrm{E}$ (see figure). The lighthouse is located at the shoreline. Find the distance $d$ from the boatto the shoreline.
The Leaning Tower of Pisa in Italy leans because it was built on unstable soil-a mixture of clay, sand, and water. The tower is approximately 58.36 meters tall from its foundation. The top of the tower leans about 5.45 meters off center. (See figure.)(a) Find the angle of lean $\alpha$ of the tower.(b) Write $\beta$ as a function of $d$ and $\theta,$ where $\theta$ is the angle of elevation to the sun.(c) Use the Law of Sines to write an equation for the length $d$ of the shadow cast by the tower in terms of $\theta$(d) Use a graphing utility to complete the table.
In the figure, $\alpha$ and $\beta$ are positive angles. (a) Write $\alpha$ as a function of $\beta$(b) Use a graphing utility to graph the function. Determine its domain and range.(c) Use the result of part (a) to write $c$ as a function of $\beta$(d) Use the graphing utility to graph the function in part (c). Determine its domain and range.(e) Use the graphing utility to complete the table. What can you conclude?
Determine whether the statement is true or false. Justify your answer.If any three sides or angles of an oblique triangle are known, then the triangle can be solved.
Determine whether the statement is true or false. Justify your answer.If a triangle contains an obtuse angle, then it must be oblique.
Determine whether the statement is true or false. Justify your answer.Two angles and one side of a triangle do not necessarily determine a unique triangle.
Can the Law of sines be used to solve a right triangle? If so, write a short paragraph explaining how to use the Law of sines to solve the following triangle. Is there an easier way to solve the triangle? Explain.$B=50^{\circ}, \quad C=90^{\circ}, \quad a=10$
Given $A=36^{\circ}$ and $a=5,$ find values of $b$ such that the triangle has (a) one solution, (b) two solutions, and (c) no solution.
In the figure, a triangle is to be formed by drawing a line segment of length $a$ from (4,3) to the positive $x$ -axis. For what value(s) of $a$ can you form(a) one triangle, (b) two triangles, and (c) no triangles? Explain your reasoning.
Use the given values to find the values of the remaining four trigonometric functions of $\theta$$\cos \theta=\frac{5}{13}, \quad \sin \theta=-\frac{12}{13}$
Use the given values to find the values of the remaining four trigonometric functions of $\theta$$\tan \theta=-\frac{8}{15}, \quad \csc \theta=\frac{17}{8}$
Write the product as a sum or difference.$6 \sin 8 \theta \cos 3 \theta$
Write the product as a sum or difference.$2 \cos 2 \theta \cos 5 \theta$
Write the product as a sum or difference.$\frac{1}{3} \cos \frac{\pi}{6} \sin \frac{5 \pi}{3}$
Write the product as a sum or difference.$\frac{5}{2} \sin \frac{3 \pi}{4} \sin \frac{5 \pi}{6}$