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In this module, we discuss trigonometric values and angles. In particular, we shall learn about two very useful algorithms which help us to find (i) value of trigonometric function when angle is given and (ii) angles when value of trigonometric function is given. In addition, we shall go through various trigonometric equations and identities. We are expected to be already familiar with them. For this reason, solutions of equations and identities are presented here without deduction and are included for reference purpose.

Values of trigonometric function

It is sufficient to know values of trigonometric functions for angles in first quarter. These angles are called acute angles (angle value less than π/2). Here, we develop algorithm, which converts angles in other quadrants in terms of acute angles. Basic idea is that angles can be expressed in terms of combination of acute angle and reference angles like 0, π/2, π and 2π. These angles demark quadrants. Using certain procedure, we can find value of trigonometric function of any angle provided we know the trigonometric value of corresponding acute angle. For the sake of convenience, we shall concentrate on acute angles π/6, π/4 and π/3, whose trigonometric function values are known to us. We follow an algorithm to determine trigonometric values as given here :

1 : Express given angle as sum or difference of acute angle and reference angles 0, π/2, π and 2π.

2 : Write trigonometric function of sum or difference as trigonometric function of acute angle. A trigonometric sum/difference combination of angles involving angles of 0, π and 2π does not change the function. However, a combination involving π/2 changes function from sine to cosine and vice-versa, tangent to cotangent and vice-versa and cosecant to secant and vice-versa.

3 : Apply sign before trigonometric function determined as above in accordance with the sign rule of trigonometric function.

f r + a = + o r g a

where “f” and “g” denote trigonometric functions, “r” denotes reference angles like 0, π/2, π and 2π and “a” denotes acute angle.

Trigonometric sign diagram

Signs of six trigonometric functions in different quadrants.

Let us consider an angle 7π/6. We are required to find sine and cotangent values of this angle. Here, we see that 7π/6 is greater than π. Hence, it is equal to π plus some acute angle, say, x.

π + x = 7 π 6 x = 7 π 6 - x = π 6 sin 7 π 6 = sin π + π 6

Since combination involves angle π, the sine of given angle retains the trigonometric function form. However, angle 7π/6 falls in third quadrant, in which sine is negative. Thus,

sin 7 π 6 = sin π + π 6 = - sin π 6 = - 3 2


cot 7 π 6 = cot π + π 6 = cot π 6 = 1 3

This method is very helpful to determine value of trigonometric function provided we know the value of trigonometric function of corresponding acute angle resulting from combination involving angles 0, π/2, π and 2π. Here, we shall work out few standard identities involving combination of angles with reference angles. We need not remember these identities. Rather, we should rely on the procedure discussed here as all of these can be derived on spot easily.

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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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.
yes that's correct
I think
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
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 .?
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
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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