<< Chapter < Page Chapter >> Page >

Explain the relationship between the Pythagorean Theorem and the Law of Cosines.

When must you use the Law of Cosines instead of the Pythagorean Theorem?

The Law of Cosines must be used for any oblique (non-right) triangle.

Algebraic

For the following exercises, assume α is opposite side a , β is opposite side b , and γ is opposite side c . If possible, solve each triangle for the unknown side. Round to the nearest tenth.

γ = 41.2° , a = 2.49 , b = 3.13

α = 120° , b = 6 , c = 7

11.3

β = 58.7° , a = 10.6 , c = 15.7

γ = 115° , a = 18 , b = 23

34.7

α = 119° , a = 26 , b = 14

γ = 113° , b = 10 , c = 32

26.7

β = 67° , a = 49 , b = 38

α = 43.1° , a = 184.2 , b = 242.8

257.4

α = 36.6° , a = 186.2 , b = 242.2

β = 50° , a = 105 , b = 45

not possible

For the following exercises, use the Law of Cosines to solve for the missing angle of the oblique triangle. Round to the nearest tenth.

a = 42 , b = 19 , c = 30 ; find angle A .

a = 14 ,   b = 13 ,   c = 20 ; find angle C .

95.5°

a = 16 , b = 31 , c = 20 ; find angle B .

a = 13 , b = 22 , c = 28 ; find angle A .

26.9°

a = 108 , b = 132 , c = 160 ; find angle C .

For the following exercises, solve the triangle. Round to the nearest tenth.

A = 35° , b = 8 , c = 11

B 45.9° , C 99.1° , a 6.4

B = 88° , a = 4.4 , c = 5.2

C = 121° , a = 21 , b = 37

A 20.6° , B 38.4° , c 51.1

a = 13 , b = 11 , c = 15

a = 3.1 , b = 3.5 , c = 5

A 37.8° , B 43.8 , C 98.4°

a = 51 , b = 25 , c = 29

For the following exercises, use Heron’s formula to find the area of the triangle. Round to the nearest hundredth.

Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. Round to the nearest tenth.

177.56 in 2

Find the area of a triangle with sides of length 20 cm, 26 cm, and 37 cm. Round to the nearest tenth.

a = 1 2 m , b = 1 3 m , c = 1 4 m

0.04 m 2

a = 12.4  ft ,   b = 13.7  ft ,   c = 20.2  ft

a = 1.6  yd ,   b = 2.6  yd ,   c = 4.1  yd

0.91 yd 2

Graphical

For the following exercises, find the length of side x . Round to the nearest tenth.

A triangle. One angle is 72 degrees, with opposite side = x. The other two sides are 5 and 6.5.
A triangle. One angle is 42 degrees with opposite side = x. The other two sides are 4.5 and 3.4.

3.0

A triangle. One angle is 40 degrees with opposite side = 15. The other two sides are 12 and x.
A triangle. One angle is 65 degrees with opposite side = x. The other two sides are 30 and 23.

29.1

A triangle. One angle is 50 degrees with opposite side = x. The other two sides are 225 and 305.
A triangle. One angle is 123 degrees with opposite side = x. The other two sides are 1/5 and 1/3.

0.5

For the following exercises, find the measurement of angle A .

A triangle. Angle A is opposite a side of length 2.3. The other two sides are 1.5 and 2.5.
A triangle. Angle A is opposite a side of length 125. The other two sides are 115 and 100.

70.7°

A triangle. Angle A is opposite a side of length 6.8. The other two sides are 4.3 and 8.2.
A triangle. Angle A is opposite a side of length 40.6. The other two sides are 38.7 and 23.3.

77.4°

Find the measure of each angle in the triangle shown in [link] . Round to the nearest tenth.

A triangle A B C. Angle A is opposite a side of length 10, angle B is opposite a side of length 12, and angle C is opposite a side of length 7.

For the following exercises, solve for the unknown side. Round to the nearest tenth.

A triangle. One angle is 60 degrees with opposite side unknown. The other two sides are 20 and 28.

25.0

A triangle. One angle is 30 degrees with opposite side unknown. The other two sides are 16 and 10.
A triangle. One angle is 22 degrees with opposite side unknown. The other two sides are 20 and 13.

9.3

A triangle. One angle is 88 degrees with opposite side = 9. Another side is 5.

For the following exercises, find the area of the triangle. Round to the nearest hundredth.

A triangle with sides 8, 12, and 17. Angles unknown.

43.52

A triangle with sides 50, 22, and 36. Angles unknown.
A triangle with sides 1.9, 2.6, and 4.3. Angles unknown.

1.41

A triangle with sides 8.9, 12.5, and 16.2. Angles unknown.
A triangle with sides 1/2, 2/3, and 3/5. Angles unknown.

0.14

Extensions

A parallelogram has sides of length 16 units and 10 units. The shorter diagonal is 12 units. Find the measure of the longer diagonal.

The sides of a parallelogram are 11 feet and 17 feet. The longer diagonal is 22 feet. Find the length of the shorter diagonal.

18.3

The sides of a parallelogram are 28 centimeters and 40 centimeters. The measure of the larger angle is 100°. Find the length of the shorter diagonal.

A regular octagon is inscribed in a circle with a radius of 8 inches. (See [link] .) Find the perimeter of the octagon.

An octagon inscribed in a circle.

48.98

A regular pentagon is inscribed in a circle of radius 12 cm. (See [link] .) Find the perimeter of the pentagon. Round to the nearest tenth of a centimeter.

A pentagon inscribed in a circle.

For the following exercises, suppose that x 2 = 25 + 36 60 cos ( 52 ) represents the relationship of three sides of a triangle and the cosine of an angle.

Draw the triangle.

A triangle. One angle is 52 degrees with opposite side = x. The other two sides are 5 and 6.

Find the length of the third side.

For the following exercises, find the area of the triangle.

A triangle. One angle is 22 degrees with opposite side = 3.4. Another side is 5.3.

7.62

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Essential precalculus, part 2' conversation and receive update notifications?

Ask