# 11.1 Graphs, trigonometric identities, and solving trigonometric  (Page 12/12)

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The other cases can be proved in an identical manner.

Find $\stackrel{^}{\mathrm{A}}$ :

1. $\begin{array}{ccc}\hfill {a}^{2}& =& {b}^{2}+{c}^{2}-2bdcos\stackrel{^}{\mathrm{A}}\hfill \\ \hfill \therefore \phantom{\rule{1.em}{0ex}}cos\stackrel{^}{\mathrm{A}}& =& \frac{{b}^{2}+{c}^{2}-{a}^{2}}{2bc}\hfill \\ & =& \frac{{8}^{2}+{5}^{2}-{7}^{2}}{2·8·5}\hfill \\ & =& 0,5\hfill \\ \hfill \therefore \phantom{\rule{1.em}{0ex}}\stackrel{^}{\mathrm{A}}& =& arccos0,5={60}^{\circ }\hfill \end{array}$

## The cosine rule

1. Solve the following triangles i.e.  find all unknown sides and angles
1. $▵$ ABC in which $\stackrel{^}{\mathrm{A}}={70}^{\circ }$ ; $b=4$ and $c=9$
2. $▵$ XYZ in which $\stackrel{^}{\mathrm{Y}}={112}^{\circ }$ ; $x=2$ 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 $\stackrel{^}{\mathrm{H}}={130}^{\circ }$ ; 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. $\stackrel{^}{\mathrm{X}}=71,{4}^{\circ }$ ; $y=3,42$  km and $z=4,03$  km
2. ; $x=103,2$  cm; $\stackrel{^}{\mathrm{Y}}=20,{8}^{\circ }$ 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 $▵DEF$ , the area is given by: $A=\frac{1}{2}DE·EFsin\stackrel{^}{E}=\frac{1}{2}EF·FDsin\stackrel{^}{F}=\frac{1}{2}FD·DEsin\stackrel{^}{D}$

In order show that this is true for all triangles, consider $▵ABC$ .

The area of any triangle is half the product of the base and the perpendicular height. For $▵ABC$ , this is: $A=\frac{1}{2}c·h.$ However, $h$ can be written in terms of $\stackrel{^}{A}$ as: $h=bsin\stackrel{^}{A}$ So, the area of $▵ABC$ is: $A=\frac{1}{2}c·bsin\stackrel{^}{A}.$

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 $\stackrel{^}{\mathrm{C}}=\stackrel{^}{\mathrm{B}}={50}^{\circ }$ . Hence $\stackrel{^}{\mathrm{A}}={180}^{\circ }-{50}^{\circ }-{50}^{\circ }={80}^{\circ }$ . Now we can use the area rule to find the area:

$\begin{array}{ccc}\hfill A& =& \frac{1}{2}cbsin\stackrel{^}{\mathrm{A}}\hfill \\ & =& \frac{1}{2}·7·7·sin{80}^{\circ }\hfill \\ & =& 24,13\hfill \end{array}$

## The area rule

Draw sketches of the figures you use in this exercise.

1. Find the area of $▵$ PQR in which:
1. $\stackrel{^}{\mathrm{P}}={40}^{\circ }$ ; $q=9$ and $r=25$
2. $\stackrel{^}{\mathrm{Q}}={30}^{\circ }$ ; $r=10$ and $p=7$
3. $\stackrel{^}{\mathrm{R}}={110}^{\circ }$ ; $p=8$ and $q=9$
2. Find the area of:
1. $▵$ XYZ with XY $=6$  cm; XZ $=7$  cm and $\stackrel{^}{\mathrm{Z}}={28}^{\circ }$
2. $▵$ PQR with PR $=52$  cm; PQ $=29$  cm and $\stackrel{^}{\mathrm{P}}=58,{9}^{\circ }$
3. $▵$ EFG with FG $=2,5$  cm; EG $=7,9$  cm and $\stackrel{^}{\mathrm{G}}={125}^{\circ }$
3. Determine the area of a parallelogram in which two adjacent sides are 10 cm and 13 cm and the angle between them is ${55}^{\circ }$ .
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 $\stackrel{^}{\mathrm{C}}$ ?

## Summary of the trigonometric rules and identities

 Pythagorean Identity Ratio Identity ${cos}^{2}\theta +{sin}^{2}\theta =1$ $tan\theta =\frac{sin\theta }{cos\theta }$
 Odd/Even Identities Periodicity Identities Cofunction Identities $sin\left(-\theta \right)=-sin\theta$ $sin\left(\theta ±{360}^{\circ }\right)=sin\theta$ $sin\left({90}^{\circ }-\theta \right)=cos\theta$ $cos\left(-\theta \right)=cos\theta$ $cos\left(\theta ±{360}^{\circ }\right)=cos\theta$ $cos\left({90}^{\circ }-\theta \right)=sin\theta$ Sine Rule Area Rule Cosine Rule $\mathrm{Area}=\frac{1}{2}\mathrm{bc}cos\mathrm{A}$ ${a}^{2}={b}^{2}+{c}^{2}-2bccosA$ $\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}$ $\mathrm{Area}=\frac{1}{2}\mathrm{ac}cos\mathrm{B}$ ${b}^{2}={a}^{2}+{c}^{2}-2accosB$ $\mathrm{Area}=\frac{1}{2}\mathrm{ab}cos\mathrm{C}$ ${c}^{2}={a}^{2}+{b}^{2}-2abcosC$

## 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 $\stackrel{^}{\mathrm{P}}=\stackrel{^}{\mathrm{Q}}={50}^{\circ }$ . 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 $\parallel$ XZ) with WX $=3$  m; YZ $=1,5$  m; $\stackrel{^}{\mathrm{Z}}={120}^{\circ }$ and $\stackrel{^}{\mathrm{W}}={30}^{\circ }$
1. Determine the distances XZ and XY.
2. Find the angle $\stackrel{^}{\mathrm{C}}$ .
3. On a flight from Johannesburg to Cape Town, the pilot discovers that he has been flying ${3}^{\circ }$ 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 $\parallel$ CD). AB $=x$ ; $\mathrm{B}\stackrel{^}{\mathrm{A}}\mathrm{D}=\mathrm{a}$ ; $\mathrm{B}\stackrel{^}{\mathrm{C}}\mathrm{D}=\mathrm{b}$ and $\mathrm{B}\stackrel{^}{\mathrm{D}}\mathrm{C}=\mathrm{c}$ . 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 $\mathrm{X}\stackrel{^}{\mathrm{Y}}\mathrm{Z}$ .
1. If XY $=x$ and $\mathrm{X}\stackrel{^}{\mathrm{Y}}\mathrm{Z}=\theta$ , show that XZ $=x\sqrt{2\left(1-cos\theta \right)}$ .
2. Calculate XZ (to the nearest kilometre) if $x=240$  km and $\theta ={132}^{\circ }$ .
6. Find the area of WXYZ (to two decimal places):
7. Find the area of the shaded triangle in terms of $x$ , $\alpha$ , $\beta$ , $\theta$ and $\phi$ :

where we get a research paper on Nano chemistry....?
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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Alexandre
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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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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
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