11.1.2: Homework

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Reading Questions

What is an oblique triangle?
State the Law of Sines.
To use the Law of Sines to solve for a missing side or angle, what information must you already have about the triangle?
What does it mean to "solve a triangle"?
When using the Law of Sines to find an unknown angle, why must you be careful?
What is the "ambiguous case" for the Law of Sines? What given information (e.g., SSS, ASA, SSA) leads to this case?
If you use the Law of Sines to find that a missing angle $\theta$ could be 40°, what is the other possible value for $\theta$?
How do you determine if both possible angles in the ambiguous case lead to valid triangles?
If your calculator returns an error when you try to use the arcsin function to find a missing angle, what does this imply about the triangle?
Why is it better to use the given values in calculations rather than values you have already calculated and rounded?

Homework

Concept Check

Can we use the Law of Sines to solve a right triangle?
Explain why we cannot use the Law of Sines to solve the triangle with $a = 8$, $b = c$, and $C = 35^{\circ}$.
Linda says, "I'm thinking of an angle whose sine is $0.3420$ (rounded to four decimal places)." Matt says, "The angle must be $20^{\circ}$ (rounded to the nearest degree)." Is he correct? Why or why not?
Sketch two possible triangles with $A = 25^{\circ}$, $b = 18$, and $a = 10$.
Try to sketch a triangle with $A = 65^{\circ}$, $b = 18$, and $a = 10$. What went wrong?

Basic Skills

For the following exercises, find the indicated side. Round to two decimal places.

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For the following exercises, find the indicated angle. Round to two decimal places.

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For the following exercises, sketch the triangle and solve. Round answers to two decimal places.

$b=7$, $A=23^{\circ}$, $B=42^{\circ}$
$c=34$, $A=53^{\circ}$, $C=26^{\circ}$
$a=1.8$, $c=2.1$, $C=44^{\circ}$
$b=8.5$, $c=6.8$, $B=23^{\circ}$
$c=75$, $A=35^{\circ}$, $B=46^{\circ}$
$a=94$, $B=29^{\circ}$, $C=84^{\circ}$
In $\triangle A B C$, $\angle A=30^{\circ}$ and $c=12$. How many triangles are possible for each of the following lengths for side $a$? Sketch the solutions in each case.
1. $a = 6$
2. $a = 4$
3. $a = 9$
4. $a = 15$

For the following exercises, find the remaining angles of the triangle (or triangles, if this is the ambiguous case of the Law of Sines). Round answers to two decimal places.

$a=66$, $c=43$, $C=25^{\circ}$
$b=10$, $c=14$, $B=20^{\circ}$
$b=100$, $c=80$, $B=49^{\circ}$
$b=4.7$, $c=6.3$, $C=54^{\circ}$
1. Sketch a triangle with $A=25^{\circ}$, $B=35^{\circ}$, and $b=16$.
2. Use the Law of Sines to find $a$.
3. Use the Law of Sines to find $c$.
4. Find $c$ without using the Law of Sines. (Hint: Sketch the altitude, $h$, from $C$ to make two right triangles. Find $h$, then use $h$ to find $c$.)

Synthesis Questions

In this problem, we show that there are two different triangles $\triangle A B C$ with $A=30^{\circ}$, $a=2$, and $c=3$.
1. Use a protractor to draw an angle $A=30^{\circ}$. Mark point $B$ on one side of the angle so that $\overline{A B}$ is 3 inches long.
2. Locate two distinct points on the other side of the angle that are each 2 inches from point $B$. These points are both possible locations for point $C$.
3. Use the Law of Sines to find two distinct possible measures for $\angle C$.
Consider the triangle $\triangle ABC$ shown below.

$Screen Shot 2022-10-06 at 6.44.08 PM.png$
1. Express the length of the altitude in terms of $\angle A$ and $c$.
2. Suppose we keep $\angle A$ and side $c$ fixed but allow $a$ to vary in length. What is the smallest value $a$ can have and still be long enough to make a triangle?
3. What are the largest and smallest values that $a$ can have in order to produce two distinct triangles $\triangle A B C$ (without changing $\angle A$ and side $c$)
For the triangle in the previous problem, suppose $A=40^{\circ}$ and $c=8$.
1. Sketch and solve the triangle if $a=12$.
2. Sketch and solve the triangle if $a=6$.
3. Sketch and solve the triangle if $a=4$.
4. For what value of $a$ is $c$ the hypotenuse of a right triangle?
For the triangle in Problem 31, suppose $A=70^{\circ}$ and $c=20$.
1. For what value of $a$ is the triangle a right triangle?
2. For what values of $a$ are there two solutions for the triangle?
3. For what values of $a$ is there one obtuse solution for the triangle?
4. For what value of $a$ is there no solution?
1. Sketch a triangle with $A=75^{\circ}$, $a=15$, and $b=6$.
2. Use the Law of Sines to find $c$.
3. Find $c$ without using the Law of Sines.
Here is a method for solving certain oblique triangles by dividing them into two right triangles. In the triangle shown, we know two angles, $A$ and $B$, and the side opposite one of them, say $a$. We would like to find side $b$.

$Screen Shot 2022-10-06 at 6.55.32 PM.png$
1. Draw the altitude $h$ from angle $C$.
2. Write an expression for $b$ in terms of $h$ and angle $A$.
3. Write an expression for $h$ in terms of angle $B$.
4. Substitute your expression for $h$ into your expression for $b$.
5. Which of the following is equivalent to the formula you wrote in part (d)?
  1. $a \sin\left( A \right)=b \sin\left( B \right)$
  2. $\frac{a}{\sin\left( A \right)}=\frac{b}{\sin\left( B \right)}$
  3. $\frac{a}{\sin\left( B \right)}=\frac{b}{\sin\left( A \right)}$

Proofs

For the following exercises, you will prove the Law of Sines using the trigonometric formula for the area of a triangle. We will derive this formul later. For now, you can use the fact that the area of triangle $ \triangle ABC $ is $ \frac{1}{2}a b \sin\left( \theta \right) $, where $ \theta $ is the angle between sides $ a $ and $ b $.

Sketch a triangle with angles $A$, $B$, and $C$, and opposite sides of lengths $a$, $b$, and $c$, respectively.
1. Write the area of the triangle in terms of $a$, $b$, and angle $C$.
2. Write the area of the triangle in terms of $a$, $c$, and angle $B$.
3. Write the area of the triangle in terms of $b$, $c$, and angle $A$.
Equate the three different expressions from Problem 36 for the area of the triangle. Multiply through by $\frac{2}{abc}$ and simplify to deduce the Law of Sines.

Applications

For the following exercises,
(a) Sketch and label a triangle to illustrate the problem.
(b) Solve the problem.

Phuong wants to know the height of a cliff on the other side of a ravine. The angle of elevation from the edge of the ravine to the cliff top is $84.6^{\circ}$. When she moves 30 feet back from the ravine, the angle of elevation is $82.5^{\circ}$. How tall is the cliff?
Nam wants to know the height of a tree in the median strip of a highway. The angle of elevation from the highway shoulder to the treetop is $43.5^{\circ}$. When he moves 10 feet farther away from the tree, the angle of elevation is $37.2^{\circ}$. How tall is the tree?
Mike and Loi are 10 kilometers apart, observing a satellite that passes directly over their heads. At a moment when the satellite is between them, Loi measures its angle of elevation as $84.6^{\circ}$, and Mike measures an angle of $87^{\circ}$. How far is the satellite from Mike?
Paige rows her kayak due east. When she began, she spotted a lighthouse 2000 meters in the distance at an angle of $14^{\circ}$ south of east. After traveling for an hour, the lighthouse was at an angle of $83^{\circ}$ south of east. How far did Paige travel, and what was her average speed?
Ivan is hiking along a straight path but needs to detour around a large pond. He turns $23^{\circ}$ from his path until clear of the pond, then walks back to his original path, intercepting it at an angle of $29^{\circ}$ and at a distance of 2 miles from where he had left the path. How far did Ivan walk in each of the two segments of his detour, and how much farther did his detour require compared with a straight line through the pond?

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Wyatt is flying to Monterey but must change course to avoid a storm. He flies $19^{\circ}$ off from his original direction until he clears the storm, then turns again to return to his original flight path, intercepting it at an angle of $54.9^{\circ}$ and at a distance of 50 miles from where he had left it. How much farther did his detour require compared with his original course?

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Geologists find an outcropping from an underground rock formation that usually indicates the presence of oil. The outcropping is on a hillside, and the formation dips another $17^{\circ}$ from the surface. If an oil well is placed 1000 meters downhill from the outcropping, how far will they drill before reaching the formation?

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A proposed ski lift will rise from a point near the base of the slope with an angle of $27^{\circ}$. At a distance of 400 meters further from the hill, the angle of elevation to the top of the ski lift is $19^{\circ}$. How long is the ski lift?

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Nhat wants to measure the height of a hill. She first plants a 50-foot-tall antenna at the hill’s peak. Then, she descends the hill and finds a point where she can see the antenna's top and bottom. The angle of elevation to the bottom of the antenna is $23^{\circ}$, and the angle of elevation to the top of the antenna is $24^{\circ}$.

$Screen Shot 2022-10-06 at 6.28.53 PM.png$
1. Find $\angle ACB$.
2. Find $\angle CAB$ at the top of the antenna.
3. How long is $BC$, the distance from the bottom of the antenna to $C$?
4. How tall is the hill?
Jorge and Camille are 1000 yards apart. The angle Jorge sees between Camille and a particular tree is $38^{\circ}$. If the tree is 800 yards from Camille, how far is it from Jorge? (There are two possible answers.)
From the lookout point on Fabrick Rock, Sang can not only see the famous "Crooked Spire" in Chesterfield, which is 8 miles away but also the red phone box in the village of Alton. Chesterfield and Alton are 7 miles apart. Fabrick Rock has a plaque that shows directions to famous sites, and from the plaque, Sang determines that the angle between the lines to the spire and the phone box measures $19^{\circ}$. How far is Fabrick Rock from the phone box? (There are two possible answers.)
A billboard of California’s gubernatorial candidate, Min, is located on the roof of a building. At a distance of 180 feet from the building, the angles of elevation to the bottom and top of the billboard are respectively $39.8^{\circ}$ and $47.3^{\circ}$. How tall is the billboard?

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For the following exercises, compute the distances in Astronomical Units. Then convert to kilometers, using the fact that 1 AU $\approx 1.5 \times 108$ km.

When observed from opposite sides of Earth's orbit, the star Alpha Centauri has a parallax of $0.76^{\prime \prime}$. How far from the Sun is Alpha Centauri?
How far from the Sun is Barnard's star, which has a parallax of $1.1^{\prime \prime}$ when observed at opposite ends of Earth's orbit?
How far from the Sun is Tau Ceti, which has a parallax of $0.55^{\prime \prime}$ when observed from opposite ends of Earth's orbit?
How far from the Sun is Sirius, which has a parallax of $0.75^{\prime \prime}$ when observed from opposite ends of Earth's orbit?

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