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The other cases can be proved in an identical manner.

Find A ^ :

  1. a 2 = b 2 + c 2 - 2 b d cos A ^ cos A ^ = b 2 + c 2 - a 2 2 b c = 8 2 + 5 2 - 7 2 2 · 8 · 5 = 0 , 5 A ^ = arccos 0 , 5 = 60

The cosine rule

  1. Solve the following triangles i.e.  find all unknown sides and angles
    1. ABC in which A ^ = 70 ; b = 4 and c = 9
    2. XYZ in which Y ^ = 112 ; x = 2 and y = 3
    3. RST in which RS = 2 ; ST = 3 and RT = 5
    4. KLM in which KL = 5 ; LM = 10 and KM = 7
    5. JHK in which H ^ = 130 ; JH = 13 and HK = 8
    6. DEF in which d = 4 ; e = 5 and f = 7
  2. Find the length of the third side of the XYZ where:
    1. X ^ = 71 , 4 ; y = 3 , 42  km and z = 4 , 03  km
    2. ; x = 103 , 2  cm; Y ^ = 20 , 8 and z = 44 , 59  cm
  3. Determine the largest angle in:
    1. JHK in which JH = 6 ; HK = 4 and JK = 3
    2. PQR where p = 50 ; q = 70 and r = 60

The area rule

The Area Rule

The area rule applies to any triangle and states that the area of a triangle is given by half the product of any two sides with the sine of the angle between them.

That means that in the D E F , the area is given by: A = 1 2 D E · E F sin E ^ = 1 2 E F · F D sin F ^ = 1 2 F D · D E sin D ^

In order show that this is true for all triangles, consider A B C .

The area of any triangle is half the product of the base and the perpendicular height. For A B C , this is: A = 1 2 c · h . However, h can be written in terms of A ^ as: h = b sin A ^ So, the area of A B C is: A = 1 2 c · b sin A ^ .

Using an identical method, the area rule can be shown for the other two angles.

Find the area of ABC:

  1. ABC is isosceles, therefore AB = AC = 7 and C ^ = B ^ = 50 . Hence A ^ = 180 - 50 - 50 = 80 . Now we can use the area rule to find the area:

    A = 1 2 c b sin A ^ = 1 2 · 7 · 7 · sin 80 = 24 , 13

The area rule

Draw sketches of the figures you use in this exercise.

  1. Find the area of PQR in which:
    1. P ^ = 40 ; q = 9 and r = 25
    2. Q ^ = 30 ; r = 10 and p = 7
    3. R ^ = 110 ; p = 8 and q = 9
  2. Find the area of:
    1. XYZ with XY = 6  cm; XZ = 7  cm and Z ^ = 28
    2. PQR with PR = 52  cm; PQ = 29  cm and P ^ = 58 , 9
    3. EFG with FG = 2 , 5  cm; EG = 7 , 9  cm and G ^ = 125
  3. Determine the area of a parallelogram in which two adjacent sides are 10 cm and 13 cm and the angle between them is 55 .
  4. If the area of ABC is 5000 m 2 with a = 150  m and b = 70  m, what are the two possible sizes of C ^ ?

Summary of the trigonometric rules and identities

Pythagorean Identity Ratio Identity
cos 2 θ + sin 2 θ = 1 tan θ = sin θ cos θ
Odd/Even Identities Periodicity Identities Cofunction Identities
sin ( - θ ) = - sin θ sin ( θ ± 360 ) = sin θ sin ( 90 - θ ) = cos θ
cos ( - θ ) = cos θ cos ( θ ± 360 ) = cos θ cos ( 90 - θ ) = sin θ
Sine Rule Area Rule Cosine Rule
Area = 1 2 bc cos A a 2 = b 2 + c 2 - 2 b c cos A
sin A a = sin B b = sin C c Area = 1 2 ac cos B b 2 = a 2 + c 2 - 2 a c cos B
Area = 1 2 ab cos C c 2 = a 2 + b 2 - 2 a b cos C

Exercises

  1. Q is a ship at a point 10 km due South of another ship P. R is a lighthouse on the coast such that P ^ = Q ^ = 50 . Determine:
    1. the distance QR
    2. the shortest distance from the lighthouse to the line joining the two ships (PQ).
  2. WXYZ is a trapezium (WX XZ) with WX = 3  m; YZ = 1 , 5  m; Z ^ = 120 and W ^ = 30
    1. Determine the distances XZ and XY.
    2. Find the angle C ^ .
  3. On a flight from Johannesburg to Cape Town, the pilot discovers that he has been flying 3 off course. At this point the plane is 500 km from Johannesburg. The direct distance between Cape Town and Johannesburg airports is 1 552 km. Determine, to the nearest km:
    1. The distance the plane has to travel to get to Cape Town and hence the extra distance that the plane has had to travel due to the pilot's error.
    2. The correction, to one hundredth of a degree, to the plane's heading (or direction).
  4. ABCD is a trapezium (ie. AB CD). AB = x ; B A ^ D = a ; B C ^ D = b and B D ^ C = c . Find an expression for the length of CD in terms of x , a , b and c .
  5. A surveyor is trying to determine the distance between points X and Z. However the distance cannot be determined directly as a ridge lies between the two points. From a point Y which is equidistant from X and Z, he measures the angle X Y ^ Z .
    1. If XY = x and X Y ^ Z = θ , show that XZ = x 2 ( 1 - cos θ ) .
    2. Calculate XZ (to the nearest kilometre) if x = 240  km and θ = 132 .
  6. Find the area of WXYZ (to two decimal places):
  7. Find the area of the shaded triangle in terms of x , α , β , θ and φ :

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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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