# 3.12 Trigonometric values, equations and identities  (Page 3/3)

 Page 3 / 3

$⇒x=2\pi -\theta =2\pi -\frac{\pi }{3}=\frac{5\pi }{3}$

Problem : Find angles in [0,2π], if

$\mathrm{cot}x=\frac{1}{\sqrt{3}}$

Solution : Considering only the magnitude of numerical value, we have :

$⇒\mathrm{cot}\theta =\frac{1}{\sqrt{3}}=\mathrm{cot}\frac{\pi }{3}$

Thus, required acute angle is π/3. Now, cotangent function is positive in first and third quadrants. Looking at the value diagram, the angle in third quadrant is :

$⇒x=\pi +\theta =\pi +\frac{\pi }{3}=\frac{4\pi }{3}$

Hence angles are π/3 and 4π/3.

## Negative angles

When we consider angle as a real number entity, we need to express angles as negative angles as well. The corresponding negative angle (y) is obtained as :

$y=x-2\pi$

Thus, negative angles corresponding to 4π/3 and 5π/3 are :

$⇒y=\frac{4\pi }{3}-2\pi =-\frac{2\pi }{3}$ $⇒y=\frac{5\pi }{3}-2\pi =-\frac{\pi }{3}$

We can also find negative angle values using a separate negative value diagram (see figure). We draw negative value diagram by demarking quadrants with corresponding angles and writing angle values for negative values. We deduct “2π” from the relation for positive value diagram.

Let us consider sinx = -√3/2 again. The acute angle in first quadrant is π/3. Sine is negative in third and fourth quadrants. The angles in these quadrants are :

$y=-\pi +\theta =-\pi +\frac{\pi }{3}=-\frac{2\pi }{3}$ $y=-\theta =-\frac{\pi }{3}$

## Zeroes of sine and cosine functions

Trigonometric equations are formed by equating trigonometric functions to zero. The solutions of these equations are :

1 : $\mathrm{sin}x=0\phantom{\rule{1em}{0ex}}⇒x=n\pi ;n\in Z$

2 : $\mathrm{cos}x=0\phantom{\rule{1em}{0ex}}⇒x=\left(2n+1\right)\frac{\pi }{2};n\in Z$

## Definition of other trigonometric functions

We define other trigonometric functions in the light of zeroes of sine and cosine as listed above :

$\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x};x\ne \left(2n+1\right)\frac{\pi }{2};n\in Z$ $\mathrm{cot}x=\frac{\mathrm{cos}x}{\mathrm{sin}x};x\ne n\pi ;n\in Z$ $\mathrm{cosec}x=\frac{1}{\mathrm{sin}x};x\ne n\pi ;n\in Z$ $\mathrm{sec}x=\frac{1}{\mathrm{cos}x};x\ne \left(2n+1\right)\frac{\pi }{2};n\in Z$

## Trigonometric equations

Trigonometric function can be used to any other values as well. Solutions of such equations are given here without deduction for reference purpose. Solutions of three equations involving sine, cosine and tangent functions are listed here :

1. Sine equation

$\mathrm{sin}x=a=\mathrm{sin}y$

$x=n\pi +{\left(-1\right)}^{n}y;n\in Z$

2. Cosine equation

$\mathrm{cos}x=a=\mathrm{cos}y$

$x=2n\pi ±y;n\in Z$

3. Tangent equation

$\mathrm{tan}x=a=\mathrm{tan}y$

$x=n\pi +y;n\in Z$

In order to understand the working with trigonometric equation, let us consider an equation :

$\mathrm{sin}x=-\frac{\sqrt{3}}{2}$

As worked out earlier, -√3/2 is sine value of two angles in the interval [0, π]. Important question here is to know which angle should be used in the solution set. Here,

$⇒\mathrm{sin}\frac{4\pi }{3}=\mathrm{sin}\frac{5\pi }{3}=-\frac{\sqrt{3}}{2}$

We can write general solution using either of two values.

$x=n\pi +{\left(-1\right)}^{n}\frac{4\pi }{3};n\in Z$ $⇒x=n\pi +{\left(-1\right)}^{n}\frac{5\pi }{3};n\in Z$

The solution sets appear to be different, but are same on expansion. Conventionally, however, we use the smaller of two angles which lie in the interval [0, π]. In order to check that two series are indeed same, let us expand series from n=-4 to n=4,

For $⇒x=n\pi +{\left(-1\right)}^{n}\frac{4\pi }{3};n\in Z$

$-4\pi +\frac{4\pi }{3}=-\frac{8\pi }{3},-3\pi -\frac{4\pi }{3}=-\frac{13\pi }{3},-2\pi +\frac{4\pi }{3}=-\frac{2\pi }{3},-\pi -\frac{4\pi }{3}=-\frac{7\pi }{3},$

$0+4\pi /3=\frac{4\pi }{3},\pi -\frac{4\pi }{3}=-\frac{\pi }{3},2\pi +\frac{4\pi }{3}=\frac{10\pi }{3},3\pi -\frac{4\pi }{3}=\frac{5\pi }{3},4\pi +\frac{4\pi }{3}=\frac{16\pi }{3}$

Arranging in increasing order :

$-\frac{13\pi }{3},-\frac{8\pi }{3},-\frac{7\pi }{3},-\frac{2\pi }{3},-\frac{\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3},\frac{10\pi }{3},\frac{16\pi }{3}$

For $⇒x=n\pi +{\left(-1\right)}^{n}\frac{5\pi }{3};n\in Z$

$-4\pi +\frac{5\pi }{3}=-\frac{7\pi }{3},-3\pi -\frac{5\pi }{3}=-\frac{14\pi }{3},-2\pi +\frac{5\pi }{3}=-\frac{\pi }{3},-\pi -\frac{5\pi }{3}=-\frac{8\pi }{3},$

$0+\frac{5\pi }{3}=\frac{5\pi }{3},\pi -\frac{5\pi }{3}=-\frac{2\pi }{3},2\pi +\frac{5\pi }{3}=\frac{11\pi }{3},3\pi -\frac{5\pi }{3}=\frac{4\pi }{3},4\pi +\frac{5\pi }{3}=\frac{17\pi }{3}$

Arranging in increasing order :

$-\frac{14\pi }{3},-\frac{8\pi }{3},-\frac{7\pi }{3},-\frac{2\pi }{3},-\frac{\pi }{3},\frac{4\pi }{3},\frac{5\pi }{3},\frac{11\pi }{3},\frac{17\pi }{3}$

We see that there are common terms. There are, however, certain terms which do not appear in other series. We can though find those missing terms by evaluating some more values. For example, if we put n = 6 in the second series, then we get the missing term -13π/3. Also, putting n=5,7, we get 10π/3 and 16π/3. Thus, all missing terms in second series are obtained. Similarly, we can compute few more values in first series to find missing terms. We, therefore, conclude that both these series are equal.

Problem : Find solution of equation :

$2{\mathrm{cos}}^{2}x+3\mathrm{sin}x=0$

Solution : Our objective here is to covert equation to linear form. Here, we can not convert sine term to cosine term, but we can convert ${\mathrm{cos}}^{2}x$ in terms of ${\mathrm{sin}}^{2}x$ .

$⇒2\left(1-\mathrm{sin}{}^{2}x\right)+3\mathrm{sin}x=0$ $⇒2-2\mathrm{sin}{}^{2}x+3\mathrm{sin}x=0$ $⇒2\mathrm{sin}{}^{2}x-3\mathrm{sin}x-2=0$

It is a quadratic equation in sinx. Factoring, we have :

$⇒2\mathrm{sin}{}^{2}x+\mathrm{sin}x-4\mathrm{sin}x-2=0$ $⇒\mathrm{sin}x\left(2\mathrm{sin}x+1\right)-2\left(2\mathrm{sin}x+1\right)=0$ $⇒\left(2\mathrm{sin}x+1\right)\left(\mathrm{sin}x-2\right)=0$

Either, sinx=-1/2 or sinx = 2. But sinx can not be equal to 2. hence,

$⇒\mathrm{sin}x=-\frac{1}{2}=\mathrm{sin}\left(\pi +\frac{\pi }{6}\right)=\mathrm{sin}\left(\frac{7\pi }{6}\right)$ $⇒x=n\pi +{\left(-1\right)}^{n}\frac{7\pi }{6};\phantom{\rule{1em}{0ex}}n\in Z$

Note : We shall not work with any other examples here as purpose of this module is only to introduce general concepts of angles, identities and equations. These topics are part of separate detailed study.

## Reciprocal identities

Reciprocals are defined for values of x for which trigonometric function in the denominator is not zero.

$\mathrm{sin}x=\frac{1}{\mathrm{cosec}x};\phantom{\rule{1em}{0ex}}\mathrm{cos}x=\frac{1}{\mathrm{sec}x};\phantom{\rule{1em}{0ex}}\mathrm{tan}x=\frac{1}{\mathrm{cot}x};$ $\mathrm{cosec}x=\frac{1}{\mathrm{sin}x};\phantom{\rule{1em}{0ex}}\mathrm{sec}x=\frac{1}{\mathrm{cos}x};\phantom{\rule{1em}{0ex}}\mathrm{cot}x=\frac{1}{\mathrm{tan}x}$

## Negative angle identities

$\mathrm{cos}\left(-x\right)=\mathrm{cos}x;\phantom{\rule{1em}{0ex}}\mathrm{sin}\left(-x\right)=-\mathrm{sin}x;\phantom{\rule{1em}{0ex}}\mathrm{tan}\left(-x\right)=-\mathrm{tan}x$

## Pythagorean identities

${\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x=1;\phantom{\rule{1em}{0ex}}1+{\mathrm{tan}}^{2}x={\mathrm{sec}}^{2}x;\phantom{\rule{1em}{0ex}}1+{\mathrm{cot}}^{2}x={\mathrm{cosec}}^{2}x$

## Sum/difference identities

$\mathrm{sin}\left(x±y\right)=\mathrm{sin}x\mathrm{cos}y±\mathrm{sin}y\mathrm{cos}x$ $\mathrm{cos}\left(x±y\right)=\mathrm{cos}x\mathrm{cos}y\mp \mathrm{sin}x\mathrm{sin}y$ $\mathrm{tan}\left(x±y\right)=\mathrm{tan}sx±\mathrm{tan}y/1\mp \mathrm{tan}x\mathrm{tan}y;\phantom{\rule{1em}{0ex}}\text{x,y and (x+y) are not odd multiple of π/2}$ $\mathrm{cot}\left(x±y\right)=\mathrm{cot}x\mathrm{cot}y\mp 1/\mathrm{cot}y±\mathrm{cot}x;\phantom{\rule{1em}{0ex}}\text{x,y and (x+y) are not odd multiple of π/2}$

## Double angle identities

$\mathrm{sin}2x=2\mathrm{sin}x\mathrm{cos}x=\frac{2\mathrm{tan}x}{1+\mathrm{tan}{}^{2}x}$ $\mathrm{cos}2x={\mathrm{cos}}^{2}x-{\mathrm{sin}}^{2}x=2{\mathrm{cos}}^{2}x-1=1-2{\mathrm{sin}}^{2}x=\frac{1-{\mathrm{tan}}^{2}x}{1+{\mathrm{tan}}^{2}x}$ $\mathrm{tan}2x=\frac{2\mathrm{tan}x}{1-\mathrm{tan}{}^{2}x}$ $\mathrm{cot}2x=\frac{{\mathrm{cot}}^{2}x-1}{2\mathrm{cot}x}$

## Triple angle identities

$\mathrm{sin}3x=3\mathrm{sin}x-4{\mathrm{sin}}^{3}x$ $\mathrm{cos}3x=4{\mathrm{cos}}^{3}x-3\mathrm{cos}x$ $\mathrm{tan}3x=\frac{3\mathrm{tan}x-{\mathrm{tan}}^{3}x}{1-3{\mathrm{tan}}^{2}x}$ $\mathrm{cot}3x=\frac{3\mathrm{cot}x-{\mathrm{cot}}^{3}x}{1-3{\mathrm{cot}}^{2}x}$

## Power reduction identities

$\mathrm{sin}{}^{2}x=\frac{1-\mathrm{cos}2x}{2}$ $\mathrm{cos}{}^{2}x=\frac{1+\mathrm{cos}2x}{2}$ $\mathrm{sin}{}^{3}x=\frac{3\mathrm{sin}x-\mathrm{sin}3x}{4}$ ${\mathrm{cos}}^{3}x=\frac{\mathrm{cos}3x+3\mathrm{cos}x}{4}$

## Product to sum identities

$2\mathrm{sin}x\mathrm{cos}y=\mathrm{sin}\left(x+y\right)+\mathrm{sin}\left(x-y\right)$ $2\mathrm{cos}x\mathrm{sin}y=\mathrm{sin}\left(x+y\right)-\mathrm{sin}\left(x-y\right)$ $2\mathrm{cos}x\mathrm{cos}y=\mathrm{cos}\left(x+y\right)+\mathrm{cos}\left(x-y\right)$ $2\mathrm{sin}x\mathrm{sin}y=-\mathrm{cos}\left(x+y\right)+\mathrm{cos}\left(x-y\right)=\mathrm{cos}\left(x-y\right)-\mathrm{cos}\left(x+y\right)$

## Sum to product identities

$\mathrm{sin}x+\mathrm{sin}y=2\mathrm{sin}\frac{\left(x+y\right)}{2}\mathrm{cos}\frac{\left(x-y\right)}{2}$ $\mathrm{sin}x-\mathrm{sin}y=2\mathrm{cos}\frac{\left(x+y\right)}{2}\mathrm{sin}\frac{\left(x-y\right)}{2}$ $\mathrm{cos}x+\mathrm{cos}y=2\mathrm{cos}\frac{\left(x+y\right)}{2}\mathrm{cos}\frac{\left(x-y\right)}{2}$ $\mathrm{cos}x-\mathrm{cos}y=-2\mathrm{sin}\frac{\left(x+y\right)}{2}\mathrm{sin}\frac{\left(x-y\right)}{2}=2\mathrm{sin}\frac{\left(x+y\right)}{2}\mathrm{sin}\frac{\left(y-x\right)}{2}$

## Half angle identities

$\mathrm{sin}\frac{x}{2}=±\sqrt{\left\{\frac{\left(1-\mathrm{cos}x\right)}{2}\right\}}$ $\mathrm{cos}\frac{x}{2}=±\sqrt{\left\{\frac{\left(1+\mathrm{cos}x\right)}{2}\right\}}$ $\mathrm{tan}\frac{x}{2}=\mathrm{cosec}x-\mathrm{cot}x=±\sqrt{\left\{\frac{\left(1-\mathrm{cos}x\right)}{\left(1+\mathrm{cos}x\right)}\right\}}=\frac{\mathrm{sin}x}{1+\mathrm{cos}x}=\frac{1-\mathrm{cos}x}{\mathrm{sin}x}$ $\mathrm{cot}\frac{x}{2}=\mathrm{cosec}x+\mathrm{cot}x=±\sqrt{\left\{\frac{\left(1+\mathrm{cos}x\right)}{\left(1-\mathrm{cos}x\right)}\right\}}=\frac{\mathrm{sin}x}{1-\mathrm{cos}x}=\frac{1+\mathrm{cos}x}{\mathrm{sin}x}$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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?
Mahi
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 ?
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?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x