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0 30 45 60 90 180
cos θ 1 3 2 1 2 1 2 0 - 1
sin θ 0 1 2 1 2 3 2 1 0
tan θ 0 1 3 1 3 - 0

These values are useful when asked to solve a problem involving trig functions without using a calculator.

Each of the trigonometric functions has a reciprocal that has a special name. The three reciprocals are cosecant (or cosec), secant (or sec) and cotangent (or cot). These reciprocals are given below:

cosec θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ

We can also define these reciprocals for any right angled triangle:

cosec θ = hypotenuse opposite sec θ = hypotenuse adjacent cot θ = adjacent opposite

Find the length of x in the following triangle.

  1. In this case you have an angle ( 50 ), the opposite side and the hypotenuse.

    So you should use sin

    sin 50 = x 100
  2. x = 100 × sin 50
  3. Use the sin button on your calculator

    x = 76 . 6 m
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Find the value of θ in the following triangle.

  1. In this case you have the opposite side and the hypotenuse to the angle θ .

    So you should use tan

    tan θ = 50 100
  2. tan θ = 0 . 5
  3. Since you are finding the angle ,

    use tan - 1 on your calculator

    Don't forget to set your calculator to `deg' mode!

    θ = 26 . 6
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In the previous example we used tan - 1 . This is simply the inverse of the tan function. Sin and cos also have inverses. All this means is that we want to find the angle that makes the expression true and so we must move the tan (or sin or cos) to the other side of the equals sign and leave the angle where it is. Sometimes the reciprocal trigonometric functions are also referred to as the 'inverse trigonometric functions'. You should note, however that tan - 1 and cot are definitely NOT the same thing.

The following videos provide a summary of what you have learnt so far.

Trigonometry - 1

Khan academy video on trigonometry - 2

Finding lengths

Find the length of the sides marked with letters. Give answers correct to 2 decimal places.

Two-dimensional problems

We can use the trig functions to solve problems in two dimensions that involve right angled triangles. For example if you are given a quadrilateral and asked to find the one of the angles, you can construct a right angled triangle and use the trig functions to solve for the angle. This will become clearer after working through the following example.

Let ABCD be a trapezium with AB = 4 cm , CD = 6 cm , BC = 5 cm and AD = 5 cm . Point E on diagonal AC divides the diagonal such that AE = 3 cm . Find A B ^ C .

  1. We draw a diagram and construct right angled triangles to help us visualize the problem.
  2. We will use triangle ABE and triangle BEC to get the two angles, and then we will add these two angles together to find the angle we want.
  3. We use sin for both triangles since we have the hypotenuse and the opposite side.
  4. In triangle ABE we find:
    sin ( A B ^ E ) = opp hyp sin ( A B ^ E ) = 3 4 A B ^ E = sin - 1 ( 3 4 ) A B ^ E = 48 , 59
    We use the theorem of Pythagoras to find EC = 4 , 4 cm . In triangle BEC we find:
    sin ( C B ^ E ) = opp hyp sin ( C B ^ E ) = 4 , 4 5 A B ^ E = sin - 1 ( 4 , 4 5 ) C B ^ E = 61 , 64
  5. We add the two angles together to get: 48 , 59 + 61 , 64 = 110 , 23
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Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Damian Reply
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Introduction about quantum dots in nanotechnology
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what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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what is biological synthesis of nanoparticles
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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