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Reciprocal Identities
sin θ = 1 csc θ csc θ = 1 sin θ
cos θ = 1 sec θ sec θ = 1 cos θ
tan θ = 1 cot θ cot θ = 1 tan θ

The final set of identities is the set of quotient identities    , which define relationships among certain trigonometric functions and can be very helpful in verifying other identities. See [link] .

Quotient Identities
tan θ = sin θ cos θ cot θ = cos θ sin θ

The reciprocal and quotient identities are derived from the definitions of the basic trigonometric functions.

Summarizing trigonometric identities

The Pythagorean identities    are based on the properties of a right triangle.

cos 2 θ + sin 2 θ = 1
1 + cot 2 θ = csc 2 θ
1 + tan 2 θ = sec 2 θ

The even-odd identities    relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle.

tan ( θ ) = tan θ
cot ( θ ) = cot θ
sin ( θ ) = sin θ
csc ( θ ) = csc θ
cos ( θ ) = cos θ
sec ( θ ) = sec θ

The reciprocal identities    define reciprocals of the trigonometric functions.

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

The quotient identities    define the relationship among the trigonometric functions.

tan θ = sin θ cos θ
cot θ = cos θ sin θ

Graphing the equations of an identity

Graph both sides of the identity cot θ = 1 tan θ . In other words, on the graphing calculator, graph y = cot θ and y = 1 tan θ .

See [link] .

Graph of y = cot(theta) and y=1/tan(theta) from -2pi to 2pi. They are the same!

Given a trigonometric identity, verify that it is true.

  1. Work on one side of the equation. It is usually better to start with the more complex side, as it is easier to simplify than to build.
  2. Look for opportunities to factor expressions, square a binomial, or add fractions.
  3. Noting which functions are in the final expression, look for opportunities to use the identities and make the proper substitutions.
  4. If these steps do not yield the desired result, try converting all terms to sines and cosines.

Verifying a trigonometric identity

Verify tan θ cos θ = sin θ .

We will start on the left side, as it is the more complicated side:

tan θ cos θ = ( sin θ cos θ ) cos θ                = ( sin θ cos θ ) cos θ                = sin θ

Verify the identity csc θ cos θ tan θ = 1.

csc θ cos θ tan θ = ( 1 sin θ ) cos θ ( sin θ cos θ )                     = cos θ sin θ ( sin θ cos θ )                     = sin θ cos θ sin θ cos θ                     = 1

Verifying the equivalency using the even-odd identities

Verify the following equivalency using the even-odd identities:

( 1 + sin x ) [ 1 + sin ( x ) ] = cos 2 x

Working on the left side of the equation, we have

( 1 + sin x ) [ 1 + sin (− x ) ] = ( 1 + sin x ) ( 1 sin x ) Since sin(− x )= sin x                                        = 1 sin 2 x Difference of squares                                        = cos 2 x cos 2 x = 1 sin 2 x

Verifying a trigonometric identity involving sec 2 θ

Verify the identity sec 2 θ 1 sec 2 θ = sin 2 θ

As the left side is more complicated, let’s begin there.

sec 2 θ 1 sec 2 θ = ( tan 2 θ + 1 ) 1 sec 2 θ sec 2 θ = tan 2 θ + 1                  = tan 2 θ sec 2 θ                  = tan 2 θ ( 1 sec 2 θ )                  = tan 2 θ ( cos 2 θ ) cos 2 θ = 1 sec 2 θ                  = ( sin 2 θ cos 2 θ ) ( cos 2 θ ) tan 2 θ = sin 2 θ cos 2 θ                  = ( sin 2 θ cos 2 θ ) ( cos 2 θ )                  = sin 2 θ

There is more than one way to verify an identity. Here is another possibility. Again, we can start with the left side.

sec 2 θ 1 sec 2 θ = sec 2 θ sec 2 θ 1 sec 2 θ                   = 1 cos 2 θ                   = sin 2 θ

Questions & Answers

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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
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?
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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 ?
Bob Reply
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?
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?
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research.net
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sciencedirect big data base
Ernesto
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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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