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Section exercises

Verbal

Describe the altitude of a triangle.

The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.

Compare right triangles and oblique triangles.

When can you use the Law of Sines to find a missing angle?

When the known values are the side opposite the missing angle and another side and its opposite angle.

In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?

What type of triangle results in an ambiguous case?

A triangle with two given sides and a non-included angle.

Algebraic

For the following exercises, assume α is opposite side a , β is opposite side b , and γ is opposite side c . Solve each triangle, if possible. Round each answer to the nearest tenth.

α = 43° , γ = 69° , a = 20

α = 35° , γ = 73° , c = 20

  β = 72° , a 12.0 , b 19.9

α = 60° , β = 60° , γ = 60°

a = 4 , α = 60° , β = 100°

  γ = 20° , b 4.5 , c 1.6

b = 10 , β = 95° , γ = 30°

For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle A is opposite side a , angle B is opposite side b , and angle C is opposite side c .

Find side b when A = 37° , B = 49° , c = 5.

b 3.78

Find side a when A = 132° , C = 23° , b = 10.

Find side c when B = 37° , C = 21 , b = 23.

c 13.70

For the following exercises, assume α is opposite side a , β is opposite side b , and γ is opposite side c . Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.

α = 119° , a = 14 , b = 26

γ = 113° , b = 10 , c = 32

one triangle, α 50.3° , β 16.7° , a 26.7

b = 3.5 , c = 5.3 , γ = 80°

a = 12 , c = 17 , α = 35°

two triangles,   γ 54.3° , β 90.7° , b 20.9 or   γ 125.7° , β 19.3° , b 6.9

a = 20.5 , b = 35.0 , β = 25°

a = 7 , c = 9 , α = 43°

two triangles,   β 75.7° ,   γ 61.3° , b 9.9 or   β 18.3° , γ 118.7° , b 3.2

a = 7 , b = 3 , β = 24°

b = 13 , c = 5 , γ = 10°

two triangles, α 143.2° , β 26.8° , a 17.3 or α 16.8° , β 153.2° , a 8.3

a = 2.3 , c = 1.8 , γ = 28°

β = 119° , b = 8.2 , a = 11.3

no triangle possible

For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.

Find angle A when a = 24 , b = 5 , B = 22°.

Find angle A when a = 13 , b = 6 , B = 20°.

A 47.8° or A 132.2°

Find angle B when A = 12° , a = 2 , b = 9.

For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.

a = 5 , c = 6 , β = 35°

8.6

b = 11 , c = 8 , α = 28°

a = 32 , b = 24 , γ = 75°

370.9

a = 7.2 , b = 4.5 , γ = 43°

Graphical

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

A triangle with an angle of 50 degrees and opposite side of length 10. Another angle is 70 degrees with side opposite of length x.

12.3

A triangle with one angle = 120 degrees. Another angle is 25 degrees with side opposite = x. The side adjacent to the 25 and 120 degree angles is of length 6.
A triangle. One angle is 45 degrees with side opposite = x. Another angle is 75 degrees. The side adjacent to the 45 and 75 degree angles = 15.

12.2  

A triangle. One angle is 40 degrees with opposite side = x. Another angle is 110 degrees with side opposite = 18.
A triangle. One angle is 50 degrees with opposite = x. Another angle is 42 degrees with opposite side = 14.

16.0  

A triangle. One angle is 111 degrees with opposite side = x. Another angle is 22 degrees. The side adjacent to the 111 and 22 degree angles = 8.6.

For the following exercises, find the measure of angle x , if possible. Round to the nearest tenth.

A triangle. One angles is 98 degrees with opposite side = 10. Another angle is x degrees with opposite side = 5.

29.7°

A triangle. One angle is 37 degrees with opposite side = 11. Another angle is x degrees with opposite side = 8.
A triangle. One angle is 22 degrees with side opposite = 5. Another angle is x degrees with opposite side = 13.

x = 76.9° or  x = 103.1°

A triangle. One angle is 59 degrees with opposite side = 5.7. Another angle is x degrees with opposite side = 5.3.

Notice that x is an obtuse angle.

A triangle. One angle is 55 degrees with side opposite = 21. Another angle is x degrees with opposite side = 24.

110.6°

A triangle. One angle is 65 degrees with opposite side = 10. Another angle is x degrees with opposite side = 12.

For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.

A triangle. One angle is 93 degrees with opposite side = 32.6. Another side is 24.1.

A 39.4 ,   C 47.6 ,   B C 20.7  

A triangle. One angle is 30 degrees. The two sides adjacent to that angle are 10 and 16.
A triangle. One angle is 25 degrees. The two sides adjacent to that angle are 18 and 15

57.1

A triangle. One angle is 51 degrees with opposite side = 3.5. The other two sides are 4.5 and 2.9.
A triangle. One angle is 58 degrees with opposite side unknown. Another angle is 51 degrees with opposite side = 9. The side adjacent to the two given angles is 11.

42.0  

A triangle. One angle is 40 degrees with opposite side = 18. One of the other sides is 25.
A triangle. One angle is 115 degrees with opposite side = 50. Another angle is 30 degrees with opposite side = 30.

430.2  

Extensions

Find the radius of the circle in [link] . Round to the nearest tenth.

A triangle inscribed in a circle. Two of the legs are radii. The central angle formed by the radii is 145 degrees, and the opposite side is 3.

Find the diameter of the circle in [link] . Round to the nearest tenth.

A triangle inscribed in a circle. Two of the legs are radii. The central angle formed by the radii is 110 degrees, and the opposite side is 8.3.

10.1

Questions & Answers

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
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
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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
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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