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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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