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Given sin α = 5 8 , with θ in quadrant I, find cos ( 2 α ) .

cos ( 2 α ) = 7 32

Using the double-angle formula for cosine without exact values

Use the double-angle formula for cosine to write cos ( 6 x ) in terms of cos ( 3 x ) .

cos ( 6 x ) = cos ( 3 x + 3 x )              = cos 3 x cos 3 x sin 3 x sin 3 x              = cos 2 3 x sin 2 3 x

Using double-angle formulas to verify identities

Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.

Using the double-angle formulas to establish an identity

Establish the following identity using double-angle formulas:

1 + sin ( 2 θ ) = ( sin θ + cos θ ) 2

We will work on the right side of the equal sign and rewrite the expression until it matches the left side.

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

Establish the identity: cos 4 θ sin 4 θ = cos ( 2 θ ) .

cos 4 θ sin 4 θ = ( cos 2 θ + sin 2 θ ) ( cos 2 θ sin 2 θ ) = cos ( 2 θ )

Verifying a double-angle identity for tangent

Verify the identity:

tan ( 2 θ ) = 2 cot θ tan θ

In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.

tan ( 2 θ ) = 2 tan θ 1 tan 2 θ Double-angle formula             = 2 tan θ ( 1 tan θ ) ( 1 tan 2 θ ) ( 1 tan θ ) Multiply by a term that results in desired numerator .             = 2 1 tan θ tan 2 θ tan θ             = 2 cot θ tan θ Use reciprocal identity for  1 tan θ .

Verify the identity: cos ( 2 θ ) cos θ = cos 3 θ cos θ sin 2 θ .

cos ( 2 θ ) cos θ = ( cos 2 θ sin 2 θ ) cos θ = cos 3 θ cos θ sin 2 θ

Use reduction formulas to simplify an expression

The double-angle formulas can be used to derive the reduction formulas    , which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.

We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let’s begin with cos ( 2 θ ) = 1 2 sin 2 θ . Solve for sin 2 θ :

cos ( 2 θ ) = 1 2 sin 2 θ   2 sin 2 θ = 1 cos ( 2 θ )     sin 2 θ = 1 cos ( 2 θ ) 2

Next, we use the formula cos ( 2 θ ) = 2 cos 2 θ 1. Solve for cos 2 θ :

        cos ( 2 θ ) = 2 cos 2 θ 1   1 + cos ( 2 θ ) = 2 cos 2 θ 1 + cos ( 2 θ ) 2 = cos 2 θ

The last reduction formula is derived by writing tangent in terms of sine and cosine:

tan 2 θ = sin 2 θ cos 2 θ           = 1 cos ( 2 θ ) 2 1 + cos ( 2 θ ) 2 Substitute the reduction formulas.           = ( 1 cos ( 2 θ ) 2 ) ( 2 1 + cos ( 2 θ ) )           = 1 cos ( 2 θ ) 1 + cos ( 2 θ )

Reduction formulas

The reduction formulas    are summarized as follows:

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

Writing an equivalent expression not containing powers greater than 1

Write an equivalent expression for cos 4 x that does not involve any powers of sine or cosine greater than 1.

We will apply the reduction formula for cosine twice.

cos 4 x = ( cos 2 x ) 2            = ( 1 + cos ( 2 x ) 2 ) 2 Substitute reduction formula for cos 2 x .            = 1 4 ( 1 + 2 cos ( 2 x ) + cos 2 ( 2 x ) )            = 1 4 + 1 2 cos ( 2 x ) + 1 4 ( 1 + cos 2 ( 2 x ) 2 )  Substitute reduction formula for cos 2 x .            = 1 4 + 1 2 cos ( 2 x ) + 1 8 + 1 8 cos ( 4 x )            = 3 8 + 1 2 cos ( 2 x ) + 1 8 cos ( 4 x )

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
Sanket Reply
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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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