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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

are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
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I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
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what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
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scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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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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