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Evaluating trigonometric ratios is a direct process in which we make use of known values, trigonometric identities and transformations or even pre-defined trigonometric tables. The evaluation of trigonometric inequalities is somewhat inverse of this process. Consider an inequality :

tan x - 3

Clearly, we need to know “x” for which this inequality holds. As pointed out earlier, trigonometric functions are “many-one” relation. The value of “x” satisfying given inequality is not an unique interval, but a series of intervals. Incidentally, however, trigonometric function values repeat after certain “period”. So this enables us to define periodic intervals in generic manner for which trigonometric inequality holds.

Trigonometric inequality

Intervals satisfying inequality, involving tangent function.

We observe that line y = - 3 intersects tangent graph at multiple points. The sections of plots satisfying the inequality are easily identified on the graph and are shown as dark red line.

Solution of trigonometric inequality

Determination of base or fundamental interval is central to solve trigonometric inequality. The function values in this interval is repeated with a periodicity of trigonometric function. The base interval depends on the nature of trigonometric function and inequality in question. The steps to find solution of trigonometric inequality are :

1 : Convert given inequality to trigonometric equation by replacing inequality sign by equality sign.

2 : Solve resulting equation in the interval [0,2π]. There are two solutions. They are the angle values at which trigonometric function has the value which is being compared in the given inequality.

3 : Convert positive angle greater than π to equivalent negative value to account for the fact that basic interval being repeated may lie on negative side of the origin (cosine, secant and tangent function).

4 : Construct base interval between two values, keeping in mind the given inequality. It is always advantageous to draw a rough intersection of graphs of each side of given inequality.

5 : If function asymptotes (tangent, cotangent, secant and cosecant) within the interval constructed, then basic interval is limited by the angle value at which function asymptotes.

6 : Generalize solution by extending base interval with the period of the trigonometric function.

In order to understand the process, let us solve the inequality given by :

tan x - 3

This example has been selected here as it involves consideration of each step as enumerated above for finding solution of inequality. Corresponding trigonometric equation, in this case, is :

tan x = - 3

The acute angle is π/3. Further, tangent function is negative in second and fourth quarter (see sign diagram). Using value diagram in conjunction with sign diagram, solution of given equation in [0, 2π] are :

Sign and value diagram

Tangent function is negative in second and fourth quarter .

x = π - θ = π - π 3 = 2 π 3 x = 2 π - θ = 2 π - π 3 = 5 π 3

Here, second angle is greater than π. Hence, equivalent negative angle is :

y = 5 π 3 - 2 π = - π 3

Questions & Answers

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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
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
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 .?
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
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.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
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
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
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 power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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