Section 1
Law of Sines
An ____ triangle is a triangle that has no right angle.
For triangle $A B C,$ the Law of Sines is $\frac{a}{\sin A}=$ ____ $=\frac{c}{\sin C}$.
Two $\quad$ and one $\quad$ determine a unique triangle.
$$\text { The area of an oblique triangle } A B C \text { is } \frac{1}{2} b c \sin A=\frac{1}{2} a b \sin C=$$ _____.
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.CAN'T COPY THE GRAPH
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=102.4^{\circ}, \quad C=16.7^{\circ}, \quad a=21.6$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=24.3^{\circ}, \quad C=54.6^{\circ}, \quad c=2.68$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=83^{\circ} 20^{\prime}, \quad C=54.6^{\circ}, \quad c=18.1$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=5^{\circ} 40^{\prime}, \quad B=8^{\circ} 15^{\prime}, \quad b=4.8$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=35^{\circ}, \quad B=65^{\circ}, \quad c=10$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=120^{\circ}, \quad B=45^{\circ}, \quad c=16$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=55^{\circ}, \quad B=42^{\circ}, \quad c=\frac{3}{4}$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$B=28^{\circ}, \quad C=104^{\circ}, \quad a=3 \frac{5}{8}$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=36^{\circ}, \quad a=8, \quad b=5$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=60^{\circ}, \quad a=9, \quad c=7$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=145^{\circ}, \quad a=14, \quad b=4$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$A=100^{\circ}, \quad a=125, \quad c=10$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$B=15^{\circ} 30^{\prime}, \quad a=4.5, \quad b=6.8$$
Use the Law of Sines to solve the triangle. Round your answers to two decimal places.$$B=2^{\circ} 45^{\prime}, \quad b=6.2, \quad c=5.8$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=110^{\circ}, \quad a=125, \quad b=100$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=110^{\circ}, \quad a=125, \quad b=200$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=76^{\circ}, \quad a=18, \quad b=20$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=76^{\circ}, \quad a=34, \quad b=21$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=58^{\circ}, \quad a=11.4, \quad b=12.8$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=58^{\circ}, \quad a=4.5, \quad b=12.8$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=120^{\circ}, \quad a=b=25$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=120^{\circ}, \quad a=25, \quad b=24$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=45^{\circ}, \quad a=b=1$$
Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places.$$A=25^{\circ} 4^{\prime}, \quad a=9.5, \quad b=22$$
Find values for $b$ such that the triangle has (a) one solution, (b) two solutions (if possible), and (c) no solution.$$A=36^{\circ}, \quad a=5$$
Find values for $b$ such that the triangle has (a) one solution, (b) two solutions (if possible), and (c) no solution.$$A=60^{\circ}, \quad a=10$$
Find values for $b$ such that the triangle has (a) one solution, (b) two solutions (if possible), and (c) no solution.$$A=105^{\circ}, \quad a=80$$
Find values for $b$ such that the triangle has (a) one solution, (b) two solutions (if possible), and (c) no solution.$$A=132^{\circ}, \quad a=215$$
Find the area of the triangle. Round your answers to one decimal place.$$A=125^{\circ}, \quad b=9, \quad c=6$$
Find the area of the triangle. Round your answers to one decimal place.$$C=150^{\circ}, \quad a=17, \quad b=10$$
Find the area of the triangle. Round your answers to one decimal place.$$B=39^{\circ}, \quad a=25, \quad c=12$$
Find the area of the triangle. Round your answers to one decimal place.$$A=72^{\circ}, \quad b=31, \quad c=44$$
Find the area of the triangle. Round your answers to one decimal place.$$C=103^{\circ} 15^{\prime}, \quad a=16, \quad b=28$$
Find the area of the triangle. Round your answers to one decimal place.$$B=54^{\circ} 30^{\prime}, \quad a=62, \quad c=35$$
Find the area of the triangle. Round your answers to one decimal place.$$A=67^{\circ}, \quad B=43^{\circ}, \quad a=8$$
Find the area of the triangle. Round your answers to one decimal place.$$B=118^{\circ}, \quad C=29^{\circ}, \quad a=52$$
A tree grows at an angle of $4^{\circ}$ from the vertical due to prevailing winds. At a point 40 meters from the base of the tree, the angle of elevation to the top of the tree is $30^{\circ}$ (see figure).(a) Write an equation that you can use to find the height $h$ of the tree.(b) Find the height of the tree.CAN'T COPY THE FIGURE
A boat is traveling due east parallel to the shoreline at a speed of 10 miles per hour. At a given time, the bearing to a lighthouse is $\mathrm{S} 70^{\circ} \mathrm{E}$ and 15 minutes later the bearing is $S 63^{\circ} \mathrm{E}$ (see figure). The lighthouse is located at the shoreline. What is the distance from the boat to the shoreline?CAN'T COPY THE FIGURE
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. 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.CAN'T COPY THE FIGURE
A bridge is built across a small lake from a gazebo to a dock (see figure). 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 $S 74^{\circ} \mathrm{E}$ and $\mathrm{S} 28^{\circ} \mathrm{E}$, respectively. Find the distance from the gazebo to the dock.CAN'T COPY THE FIGURE
A 10-meter utility 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.CAN'T COPY THE FIGURE
A plane flies 500 kilometers with a bearing of $316^{\circ}$ from Naples to Elgin (see figure). The plane then flies 720 kilometers from Elgin to Canton (Canton is due west of Naples). Find the bearing of the flight from Elgin to Canton.CAN'T COPY THE FIGURE
The angles of elevation to an airplane from two points $A$ and $B$ on level ground are $55^{\circ}$ and $72^{\circ}$, respectively. The points $A$ and $B$ are 2.2 miles apart, and the airplane is east of both points in the same vertical plane.(a) Draw a diagram that represents the problem. Show the known quantities on the diagram.(b) Find the distance between the plane and point $B$(c) Find the altitude of the plane.(d) Find the distance the plane must travel before it is directly above point $A$
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's shadow is 16 meters long and points directly up the slope. The angle of elevation from the tip of the shadow to the sun is $20^{\circ} .$(a) Draw a diagram that represents the problem. Show the known quantities on the diagram and use a variable to indicate the height of the flagpole.(b) Write an equation that you can use to find the height of the flagpole.(c) Find the height of the flagpole.
Air traffic controllers continuously monitor the angles of elevation $\theta$ and $\phi$ to an airplane from an airport control tower and from an observation post 2 miles away (see figure). Write an equation giving the distance $d$ between the plane and the observation post in terms of $\theta$ and $\phi$.CAN'T COPY THE FIGURE
In the figure, $\alpha$ and $\beta$ are positive angles.CAN'T COPY THE GRAPH(a) Write $\alpha$ as a function of $\beta$(b) Use a graphing utility to graph the function in part (a). 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) Complete the table. What can you infer?$$\begin{array}{|l|l|l|l|l|l|l|l|}\hline \beta & 0.4 & 0.8 & 1.2 & 1.6 & 2.0 & 2.4 & 2.8 \\\hline \alpha & & & & & & & \\\hline c & & & & & & & \\\hline\end{array}$$
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.
Determine whether the statement is true or false. Justify your answer.When you know the three angles of an oblique triangle, you can solve the triangle.
Determine whether the statement is true or false. Justify your answer.The ratio of any two sides of a triangle is equal to the ratio of the sines of the opposite angles of the two sides.
Describe the error.The area of the triangle with $C=58^{\circ}, b=11$ feet, and$c=16$ feet is$\begin{aligned} \text { Area } &=\frac{1}{2}(11)(16)\left(\sin 58^{\circ}\right) \\ &=88\left(\sin 58^{\circ}\right) \end{aligned}$$=74.63$ square feet.CAN'T COPY THE FIGURE
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.CAN'T COPY THE GRAPH
Can the Law of sines be used to solve a right triangle? If so, use the Law of Sines to solve the triangle with $B=50^{\circ}, \quad C=90^{\circ}, \quad$ and $\quad a=10$Is there another way to solve the triangle? Explain.
(a) Write the area $A$ of the shaded region in the figure as a function of $\theta .$(b) Use a graphing utility to graph the function.(c) Determine the domain of the function. Explain how decreasing the length of the eight-centimeter line segment affects the area of the region and the domain of the function.CAN'T COPY THE GRAPH