<< Chapter < Page Chapter >> Page >

x = 2 π - θ = 2 π - π 3 = 5 π 3

Problem : Find angles in [0,2π], if

cot x = 1 3

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

cot θ = 1 3 = cot π 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 = π + θ = π + π 3 = 4 π 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 π

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

y = 4 π 3 - 2 π = - 2 π 3 y = 5 π 3 - 2 π = - π 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.

Trigonometric value diagram

Trigonometric value diagram for negative angles

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 = - π + θ = - π + π 3 = - 2 π 3 y = - θ = - π 3

Trigonometric equations

Zeroes of sine and cosine functions

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

1 : sin x = 0 x = n π ; n Z

2 : cos x = 0 x = 2 n + 1 π 2 ; n Z

Definition of other trigonometric functions

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

tan x = sin x cos x ; x 2 n + 1 π 2 ; n Z cot x = cos x sin x ; x n π ; n Z cosec x = 1 sin x ; x n π ; n Z sec x = 1 cos x ; x 2 n + 1 π 2 ; n 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

sin x = a = sin y

x = n π + - 1 n y ; n Z

2. Cosine equation

cos x = a = cos y

x = 2 n π ± y ; n Z

3. Tangent equation

tan x = a = tan y

x = n π + y ; n Z

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

sin x = - 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,

sin 4 π 3 = sin 5 π 3 = - 3 2

We can write general solution using either of two values.

x = n π + - 1 n 4 π 3 ; n Z x = n π + - 1 n 5 π 3 ; n 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 π + - 1 n 4 π 3 ; n Z

- 4 π + 4 π 3 = - 8 π 3 , - 3 π - 4 π 3 = - 13 π 3 , - 2 π + 4 π 3 = - 2 π 3 , - π - 4 π 3 = - 7 π 3 ,

0 + 4 π / 3 = 4 π 3 , π - 4 π 3 = - π 3 , 2 π + 4 π 3 = 10 π 3 , 3 π - 4 π 3 = 5 π 3 , 4 π + 4 π 3 = 16 π 3

Arranging in increasing order :

- 13 π 3 , - 8 π 3 , - 7 π 3 , - 2 π 3 , - π 3 , 4 π 3 , 5 π 3 , 10 π 3 , 16 π 3

For x = n π + - 1 n 5 π 3 ; n Z

- 4 π + 5 π 3 = - 7 π 3 , - 3 π - 5 π 3 = - 14 π 3 , - 2 π + 5 π 3 = - π 3 , - π - 5 π 3 = - 8 π 3 ,

0 + 5 π 3 = 5 π 3 , π - 5 π 3 = - 2 π 3 , 2 π + 5 π 3 = 11 π 3 , 3 π - 5 π 3 = 4 π 3 , 4 π + 5 π 3 = 17 π 3

Arranging in increasing order :

- 14 π 3 , - 8 π 3 , - 7 π 3 , - 2 π 3 , - π 3 , 4 π 3 , 5 π 3 , 11 π 3 , 17 π 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 cos 2 x + 3 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 cos 2 x in terms of sin 2 x .

2 1 - sin 2 x + 3 sin x = 0 2 - 2 sin 2 x + 3 sin x = 0 2 sin 2 x 3 sin x 2 = 0

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

2 sin 2 x + sin x 4 sin x 2 = 0 sin x 2 sin x + 1 2 2 sin x + 1 = 0 2 sin x + 1 sin x 2 = 0

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

sin x = - 1 2 = sin π + π 6 = sin 7 π 6 x = n π + - 1 n 7 π 6 ; n 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.

Trigonometric identities

Reciprocal identities

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

sin x = 1 cosec x ; cos x = 1 sec x ; tan x = 1 cot x ; cosec x = 1 sin x ; sec x = 1 cos x ; cot x = 1 tan x

Negative angle identities

cos - x = cos x ; sin - x = - sin x ; tan - x = - tan x

Pythagorean identities

cos 2 x + sin 2 x = 1 ; 1 + tan 2 x = sec 2 x ; 1 + cot 2 x = cosec 2 x

Sum/difference identities

sin x ± y = sin x cos y ± sin y cos x cos x ± y = cos x cos y sin x sin y tan x ± y = tan s x ± tan y / 1 tan x tan y ; x,y and (x+y) are not odd multiple of π/2 cot x ± y = cot x cot y 1 / cot y ± cot x ; x,y and (x+y) are not odd multiple of π/2

Double angle identities

sin 2 x = 2 sin x cos x = 2 tan x 1 + tan 2 x cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x = 1 - tan 2 x 1 + tan 2 x tan 2 x = 2 tan x 1 - tan 2 x cot 2 x = cot 2 x - 1 2 cot x

Triple angle identities

sin 3 x = 3 sin x 4 sin 3 x cos 3 x = 4 cos 3 x 3 cos x tan 3 x = 3 tan x tan 3 x 1 - 3 tan 2 x cot 3 x = 3 cot x cot 3 x 1 - 3 cot 2 x

Power reduction identities

sin 2 x = 1 - cos 2 x 2 cos 2 x = 1 + cos 2 x 2 sin 3 x = 3 sin x sin 3 x 4 cos 3 x = cos 3 x + 3 cos x 4

Product to sum identities

2 sin x cos y = sin x + y + sin x - y 2 cos x sin y = sin x + y - sin x - y 2 cos x cos y = cos x + y + cos x - y 2 sin x sin y = - cos x + y + cos x - y = cos x - y - cos x + y

Sum to product identities

sin x + sin y = 2 sin x + y 2 cos x - y 2 sin x - sin y = 2 cos x + y 2 sin x - y 2 cos x + cos y = 2 cos x + y 2 cos x - y 2 cos x - cos y = - 2 sin x + y 2 sin x - y 2 = 2 sin x + y 2 sin y - x 2

Half angle identities

sin x 2 = ± { 1 - cos x 2 } cos x 2 = ± { 1 + cos x 2 } tan x 2 = cosec x cot x = ± { 1 cos x 1 + cos x } = sin x 1 + cos x = 1 cos x sin x cot x 2 = cosec x + cot x = ± { 1 + cos x 1 cos x } = sin x 1 cos x = 1 + cos x sin x

Questions & Answers

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Basic mathematics review. OpenStax CNX. Jun 06, 2012 Download for free at http://cnx.org/content/col11427/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Basic mathematics review' conversation and receive update notifications?

Ask