Exercises: 3.3 The Law of Cosines

1.

1. Simplify $5^2+7^2-2(5)(7)\cos \theta$
2. Evaluate the expression in part $a$ for $\theta =29°$
3. Evaluate the expression in part $a$ for $\theta =151°$

2.

1. Simplify ${26.1}^2+{32.5}^2-2(26.1)(32.5)\cos \varphi$
2. Evaluate the expression in part $a$ for $\varphi =64°$
3. Evaluate the expression in part $a$ for $\varphi =116°$

3.

1. Solve $b^2=a^2+c^2-2ac\cos \beta$ for $\cos \beta$
2. For the equation in part $a$, find $cosbeta$ if $a=5,b=11,$ and $c=8.$

4.

1. Solve $a^2=b^2+c^2-2bc\cos \alpha$ for $\cos \alpha$
2. For the equation in part $a$, find $\cos \alpha$ if $a=4.6,b=7.2,$ and $c=9.4.$

5.

1. The equation $9^2=b^2+4^2-2b(4)\cos \alpha$ is quadratic in $b.$ Write the equation in standard form.
2. Solve the equation in part $a$ for $b$ if $\alpha =48°.$

6.

1. The equation $5^2=6^2+c^2-2(6)c\cos \beta$ is quadratic in $c.$ Write the equation in standard form.
2. Solve the equation in part $a$ for $c$ if $\beta =126°.$

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

$a=23,b=14,c=18$

18.

$a=18,b=25,c=19$

19.

$a=16.3,b=28.1,c=19.4$

20.

$a=82.3,b=22.5,c=66.8$

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

$B=47°,a=23,c=17$

36.

$C=32°,a=14,b=18$

37.

$a=8,b=7,c=9$

38.

$a=23,b=34,c=45$

39.

$b=72,c=98,B=38°$

40.

$a=28,c=41,A=27°$

41.

$c=5.7,A=59°,B=82°$

42.

$b=82,A=11°,C=42°$

43.

A surveyor would like to know the distance $PQ$ 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 $76°.$ She also finds $OP=1400$ meters and $OQ=600$ meters. Calculate the distance $PQ.$

44.

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 $41°,$ and the distances to $G$ and $H$ are 900 yards and 2500 yards, respectively. How long will the tunnel be?

45.

Two pilots leave an airport at the same time. One pilot flies $3°$ east of north at a speed of 320 miles per hour; the other flies $157°$ east of north 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?

46.

Two boats leave port at the same time. One boat sails due west at a speed of 17 miles per hour, the other powers $42°$ east of north 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?

47.

Caroline wants to fly directly south from Indianapolis to Cancún, Mexico, a distance of 1290 miles. However, to avoid bad weather, she flies for 400 miles on a heading $18°$ east of south. What is the heading to Cancún from that location, and how far is it?

48.

Alex sails 8 miles from Key West, Florida, on a heading $40°$ east of south. 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?

49.

The phone company wants to erect a cell tower on a steep hill inclined $26°$ 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?

50.

Sandstone Peak rises 3500 feet above the desert. The park service plans to run an aerial tramway up the north face, which is inclined at an angle of $68°$ 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?

51.

The sides of a triangle are 27 cm, 15 cm, and 20 cm. Find the area of the triangle. (Hint: Find one of the angles first.)

52.

The sides of a parallelogram are 10 inches and 8 inches, and form an angle of $130°.$ Find the lengths of the diagonals of the parallelogram.

53.

54.

55.

56.

57.

1. 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 $c^2=(x^2+y^2)+b^2-2bx.$

58.

Demonstrate the other two cases of the law of cosines:

• $a^2=b^2+c^2-2bc\cos A$
• $b^2=a^2+c^2-2ac\cos B$

(Hint: See Problem 57 and switch the roles of $a$ and $c,$ etc.)

59.

Use the law of cosines to prove the projection laws:
$a=b\cos C+c\cos B$
$b=c\cos A+a\cos C$
$c=a\cos B+b\cos A$
Illustrate with a sketch. (Hint: Add together two of the versions of the law of cosines.)

60.

If $\mathrm{△}ABC$ is isosceles with $a=b,$ show that $c^2=2a^2(1-\cos C).$

61.

Use the law of cosines to prove:
$1+\cos A={\displaystyle \frac{(a+b+c)(-a+b+C)}{2bc}}$
$1-\cos A={\displaystyle \frac{(a-b+c)(-a+b+C)}{2bc}}$

62.

Prove that
${\displaystyle \frac{\cos A}{a}}+{\displaystyle \frac{\cos B}{b}}+{\displaystyle \frac{\cos C}{c}}={\displaystyle \frac{a^2+b^2+c^2}{2abc}}$