Formula- By B B Susheel Kumar

8/4/2019 Formula- By B B Susheel Kumar

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TO DETERMINE THE VALUES OF OTHER TRIGNOMETRIC RATIOS WHEN ONE

TRIGNOMETRIC RATIO IS GIVEN:

If one of the t-ratio is given , the values of other t-ratios can be obtained by constructing a right angledtriangle and using the trigonometric identities given above

For acute angled traingle, we can write other t ratios in terms of given ratio:

Let sinθ=s=perp

hyp=

s

1

cosθ= = 1−sin2 ; tanθ=

sin

1−sin2; secθ=

1

1−sin2

;

cosecθ=1

sin ; cotθ= 1−sin2

sin

We can express sinθ in terms of other trigonometric functions by above method:

sinθ= 1−cos2 =

tan

1tan2

=1

cosec = sec2−1

sec= 1tan

2

tan

For ex. sinθ=1/3, since sine is +ve in Q1 and Q2(II quadrant), we have

cosθ= 1−1

9or - 1−

1

9ie.

2 23

or −2 2

3

according as θ ∈Q1 or θ ∈Q2

We can find other ratios by forming a rightangled traingle. Let tanθ=4/3, 3

2,

then since in Q3, sine and cosine both are negative, we have sinθ=-4

5; cosθ=

−3

5

TRIGNOMETRIC RATIOS OF STANDARD and QUANDRANTAL ANGLES:

Radians

0

6

4

3

2

3

2

2

12

5

12

Degrees 0 300 450 600 900 1800 2700 3600 150 750

sinθ

0

1

2

1

2 32 1 0 -1 0

3−1

2 2 31

2 2

cosθ

1

32

1

21

2 0 -1 0 1

31

2 2 3−1

2 2

tanθ

0

1

3 1

3 ∞

0

∞

0

2− 3

VALUES OF T-FUNCTIONS OF SOME FREQUENTY OCCURING ANGLES.

Radians 0 2

3

3

4

5

62n1

2

n

Degrees 1200 1350 1500

(odd )

2

(any )

sinθ 32

1

21

2

(-1)n

0

cosθ−1

2−

1

2− 3

2 0

(-1)n

tanθ − 3-1

−1

3∞

0

e.g. cos(odd

2)=0; cos( odd )=-1, cos(even ) =1

cos 2n−1

2=0, cos( 2n-1) =-1, cos(2n ) =1

sin(any ) =0, tan(any ) =0 sin n =tan n =0 if n=0,1,2

3 for http://pucpcmb.wordpress.com

1− s2

s

1

1− s2

13

4

3

5



sin

2= sin

5

2=sin

9

2=.......=1

sin(3

2) = sin

7

2= sin

11

2= ..........=-1

Some interesting results about allied angles:

1. cosn `=(-1)n , sin n =0 2)Sin(nП + θ ) =(-1)n sin θ;

cos(nП + θ )=(-1)n cos θ

3) cos(n

2+θ)=(-1)n+1/2 sinθ if n is odd 4)sin(

n

2+θ)=(-1)n-1/2 cosθ if n is odd

=(-1)n/2 sinθ if n is even =(-1)n/2 cosθ if n is even

DOMAIN AND RANGE OF TRIGNOMETRIC FUNCTIONS:

Function Domain Range

sine

cosine

tangent

cotangent

secant

cosecant

R

R

R-{(2n+1) 2

}: nε Z

R-{n }; nεZ

R-{(2n+1)

2}: nε Z

R-{n }; nεZ

[-1, 1]

[-1, 1]

R

R

(- ∞ ,-1] υ [1, ∞ )

(- ∞ ,-1] υ [1, ∞ )

ASTC RULE:(QUADRANT RULE):‘ASTC’ rule to remember the

signs ‘allied angles’

A denotes all angles are positive in the I quadrant

S says that sin (and hence cosec) is positive in the II quadrant.

The rest are negative.T means tan (and hence cot) is positive in the IIIquadrant. The rest are negative. C means cos (and hence sec) is positive

in the IV quadrant. The rest are negative.

The trignometric ratios of allied angles can be easily remembred from the

following clues:

1. First decide the sign +ve or -ve depending upon the quandrant in whichthe angle lies using QUADRANT RULE.

2. a) When the angle is 90+θ or 270-θ, the trignometric ratio changes

from sine→cosine, cosine→sine, tan→cot, cot→tan, sec→cosec,cosec→sec.

Hence the sine and cosine, tan &cot, sec & cosec are called co - ratios.

b) When the angle is 180+θ or 360 θ , -θ, the trignometrc ratio is remains the same. i.e

sin →sine, cosine→cosine , tan→tan, cot→cot, sec→sec, cosec→cosec.

ALLIED ANGLE’ FORMULAE:Trignometrc ratios of allied angles

θ sinθ cosθ tanθ secθ cosecθ cotθ

-θ -sinθ cosθ -tanθ secθ -cosecθ -cotθ

900 -θ cosθ sinθ cotθ cosecθ secθ tanθ

900 + θ cosθ -sinθ -cotθ -cosecθ secθ -tanθ

1800 -θ sinθ -cosθ -tanθ -secθ cosecθ -cotθ

1800+θ -sinθ -cosθ tanθ -secθ -cosecθ cotθ

2700 -θ -cosθ -sinθ cotθ -cosecθ -secθ tanθ

2700 +θ cosθ -sinθ -cotθ -cosecθ secθ -tanθ

3600 -θ -sinθ cosθ -tanθ secθ -cosecθ cotθ


S A

T C

90-θn.360 + θ90+θ

180-θ

180+θ

270-θ360-θ270+θ-θ



The above may be summed up as follows: Any angle can be expressed as n.90+θ where n is anyinteger and θ is an angle less than 900. To get any t. ratios of this angle

a) observe the quandrant n.90+θ lies and determine the sign (+ve or -ve).

b) If n is odd the function will change into its co function ( i.e sine↔cosine; tan↔cot; sec↔cosec. If n iseven t-ratios remains the same.(i.e sin↔sin, cos↔cos etc)

ILLUSTRATION: 1. To determine sin(540-θ), we note that 5400 -θ =6 x 900 -θ is a second quadrantangle if 0<θ<900. In this quadrant , sine is positive and since the given angle contains an even multiple of

2

, the sine function is retained . Hence sin(540- θ ) =sin θ.

2. To determine cos(6300 - θ ), we note 6300 - θ =7 x 900 - θ is a third quadrant angle if 0< θ <900. In

this quadrant cosine is negative and, since the given angle contains an odd multiple of

2, cosine is

replaced by sine. Hence cos(6300 - θ ) = -sin θ.

Short cut: Supposing we have to find the value of t- ratio of the angle θ

Step1: Find the sign of the t-ratio of θ , by finding in which quadrant the angle θ lies. This can be done

by applying the quadrant rule, i.e. ASTC Rule.

Step 2: Find the numerical value of the t-ratio of θ using the following method:

t-ratios of θ=

t- ratio of (1800- θ ) with proper sign if θ lies in the second quandrant

e.g.: cos1200 = -cos600 = -1/2

t-ratio of ( θ -180) with proper sign if θ lies in the third quandrant

e.g: sin2100 = -sin300 = -1/2

t-ratio of (360- θ ) with proper sign if θ lies in the fourth quandrant

e.g: cosec3000

= -cosec600

= −

2

3

t-ratio of θ-n (3600 ) if θ>3600

d) If θ is greater than 3600 i.e. θ =n.3600 +α , then remove the multiples of 3600 (i.e. go on subtractingfrom 3600 till you get the angle less than 3600 ) and find the t-ratio of the remaining angle by applyingthe above method. e.g: tan10350 =tan6750 (1035-360) =tan3150 = -tan450 =-1

COMPLIMENTARY AND SUPPLIMENTARY ANGLES:

If θ is any angle then the angle

2- θ is its complement angle and the angl

e - θ is its

supplement angle.

a) trigonometric ratio of any angle = Co-trigonometric ratio of its complement

sin θ = cos(90- θ ), cos θ = sin(90- θ ), tan θ = cot(90- θ ) e.g. sin600 =cos300 , tan600 =cot300 .

b) sin of(any angle) = sin of its supplement ; cos of ( any angle) = -cos of its supplement

tan of any angle = - tan of its supplement i.e. sin 300 =sin 1500 , cos 600 =-cos 1200

CO-TERMINAL ANGLES: Two angles are said to be co terminal angles , if their terminal sides

are one and the same. e.g. θ and 360+ θ or θ and n.360+ θ ; - θ and 360- θ or - θ and n.360- θ

are co terminal angles : a) Trig functions of θ and n.360+ θ are same

b) Trig functions of -θ and n.360- θ are same .

TRIGNOMETRIC RATIOS OF NEGETIVE ANGLES:For negative angles always use the following relations:

c) sin(- θ ) = -sin θ cos(- θ ) = cos θ, tan(- θ )= -tan θ , cosec(- θ )= -cosec θ ; se(- θ ) =sec θ ;

ci) cot(- θ) =sec θ(V.IMP)




TRIPLE ANGLES: T - ratios of 3 θ in terms of those of θ

Sin 3A = 3 sin A – 4 sin3A ;

cos 3A = 4 cos3A – 3 cos A ;

tan3A =3tanA−tan

3 A

1−3tan2 A

;

DEDUCTIONS:

4 sin3A =3 sin A -Sin 3A ;

sin3A =1

4( 3 sin A -Sin 3A ).

4 cos3A =3 cos A +cos 3A;

cos3A = 1

4( 3 cos A +cos 3A )

TRIGNOMETRC RATIOS OF HALFANGLES-t ratios of sub multiple angles

a) sinθ =2sin

2cos

2=

2tan

2

1tan2

2

b) cosθ=cos

2

2 -sin

2

2 =2cos

2

2 -1

=1-2sin2

2=

1−tan2

2

1tan2

2

c)tanθ=

2tan

2

1−tan2

2

DEDUCTIONS:

1+cosθ=2cos2

2; 1-cosθ=2sin2

2

1−cos

1cos =tan2

2;

1cos

1−cos=cot2

2

1−sin

1sin = tan

24 −

2 ;

1sin

1−sin = cot

2 4

2

sin

1cos =tan

2;

sin

1−cos =cot

2

cos

1sin = tan4 −

2 ;

cos

1−sin = cot4

2 Transformation formulae:

a) SUMS AND DIFFERENCE TO PRODUCT FORMULAE:

Formula that express sum or difference into products

Sin C + sin D = 2sinC D

2cos

C–D

2Sin C – sin D = 2cos

C D

2sin

C–D

2

Cos C + cos D = 2cosC D

2cos

C–D

2Cos C – cos D = 2sin

C D

2sin

D−C

2

or −2sin C D2

sin C − D2

b) PRODUCT-TO-SUM OR DIFFERENCE FORMULAE :formula which expressproducts as sum or Difference of sines and cosines.

2 sin A cos B = sin (sum) + sin (diff) i.e 2 sinA cosB = sin(A+B) + sin(A-B)

2 cos A sin B = sin (sum) – sin (diff) i.e 2 cosA sinB = sin(A+B) - sin(A-B)

2 cos A cos B = cos (sum) + cos (diff) i.e. 2 cosA.cosB = cos(A+B)+cos(A-B)

2 sin A sin B = cos (diff) – cos (sum) i.e. -2 sinA.sin B = cos(A+B)-cos(A-B)

OR 2 sinA.sin B = cos(A-B)-cos(A+B)

EXPRESSION FOR Sin(A/2) and cos(A/2) in terms of sinA:

sin A

2cos

A

2 2

=1+sinA so that sinA

2cos

A

2= ± 1 sinA

sin A

2−cos

A

2 2

=1-sinA so that sinA

2−cos

A

2= ± 1− sinA

By addition and subtraction, we have

2 sinA

2= ± 1 sinA ± ± 1− sinA ; 2 cos

A

2= ± 1 sinA ∓ ± 1− sinA

Using suitable signs , we can find sin A2 , cos A

2




VALUES OF TRIGNOMETRICAL RATIOS OF SOME IMPORTANT ANGLES:

Angle→

Ratio↓

7 1

2

0 150 180

221

2

036 0 750

sin 8−2 6−2 2

4

or 4− 6− 22 2

3−1

2 2

5−1

4

1

2 2− 2

1

4 10−2 5 31

2 2

cos 82 6−2 24

or

4 6 22 2

31

2 2

1

4 102 5

1

2 2 2

1

4 51 3−1

2 2

tan 6− 4− 3 2or

3− 2 2−1

2- 3 25−10 5

5

2−1 5−2 5 2+ 3

cot 6± 4± 3 2or

3 2 21

2+ 3 52 5 21

12

5 2- 3

sec 16−10 28 3−6 ( 6− 2 )

2−2

5 4−2 2 5−1 6 2

sin22½0 =1

2 2− 2 ;

cos22½0 =1

2 2 2 ;

tan22½0 = 2−1 ;cot22½0= 21

sin180 =1

4 5−1 =cos720 ;

cos180 =1

4 102 5 =sin720 ;

sin360 =1

4 10−2 5 =cos540 ;

cos360 =1

4 51 =sin540

tan7 ½0= 6− 4− 3 2

cot7½0= 6± 4± 3 2

sin90 = 3 5− 3− 54

cos90 = 3 5 3− 54

MAXIMUM AND MININUM VALUES :

1. since sin2A+cos2A =1, hence each of sinA and cosA is numerically less than or equal to unity, that is

|sinA|≤1 and |cosA|≤1 i.e. -1≤sinA≤1 and -1≤cosA≤1

2. Since secA and cosecA are respectively reciprocals of cosA and sinA, therefore the values of secA andcosecA are always numerically greater than or equal to unity. That is

secA≥1 or secA≤-1 and cosecA≥1 or cosecA≤-1, In otherwords w

e never have -1<cosecA<1 and-1<secA<1

3. tanA and cotA can assume any real value.

For all values of θ, -1≤sin θ≤1 and -1≤cos θ≤1a)Max . sin θ =1; Min . sin θ =-1

b)Max . (sin θ cos θ)=Max sin2

2 =1

2; Min. (sin θ cos θ) =Min sin2

2 = -1

2

4.If y =a sinx + bcosx +c, then ∀ a , b , c∈ R , we can write y=c+ a2b2 sin(x+α)

Where a= r cos α b=r sin α ⇒ r= a2b

2 tanα =b

a; since -1≤sin (x+α )≤1

∴ c- a2b

2 ≤y≤c+ a2b

2 Hence Max. (a sinx + bcosx +c) =c+ a2b

2 and

Min (a sinx + bcosx +c)= c- a2

b2




GRAPHS OF TRIGNOMETRIC FUNCTIONS

I quadrant II quadrant III quadrant IV quadrant

sinθ increases from 0 to 1 decreases from 1 to 0 decreses from 0 to -1 increases from -1 to 0

cosθ decreases from 1 to 0 decreases from 0 to -1 increases from -1 to 0 increases from 0 to 1

tanθ increases from 0 to ∞ increases from ∞ to 0 increases from 0 to ∞ increases from −∞ to 0

cotθ decreases from ∞ to 0 dec. from 0 to∞

decreases from ∞ to 0 decreases from 0 to ∞secθ increses from 1 to ∞ incr. from ∞ to -1 decreases from -1 to −∞ decreases from ∞ to 1

cosecθ decreases from ∞ to 1 increases from 1 to ∞ increases from −∞ to -1 decreases from -1 to ∞ -

Graph of sinx Graph of cosecx

Graph of cosx Graph of secx

Graph of tanx Graph of cotx




RELATION BETWEEN THE SIDES & ANGLES OF A TRIANGLE:

A traingle consists of 6 elements, three angles and three sides. The angles of traingle ABCare denoted by A,B, and C. a,b, and c are respectively the sides opposite to the angles A,Band C.

In any traingle ABC , the following results or rule hold good.

1 Sine rule’: a = 2R sin A, b = 2R sin B, c = 2R sin C iea

sinA=

b

sinB=

c

sinC =2R Where R is

the circum radius of circum circle that passes through the vertices of the traingle.

2.‘Cosine rule’: a2 =b2 +c2 -2bc cosA or cos A =b

2c

2 – a

2

2bc

b2 =a2 +c2 -2ac cosB or cos B =c2a

2 – b

2 2ca

c2 =a2 +b2 -2ab cosC or cos C =a2b

2 – c

2 2ab

3.Projection rule’:

a = b cos C + c cos B; b = c cos A + a cos C; c = a cos B + b cos A

4.Napier's formula or ‘Law of Tangents’:

tan B–C

2=[b –c

bc]cot

A

2or b−c

bc =tan

B−C

2

tanBC 2

tan A–B

2=[a –b

ab

]cotC

2or

a−b

ab

=

tanA− B

2

tan A B2

etc.

5.‘Half-angle rule’: In any traingle ABC, a+b+c =2s, where 2s is the perimeter of the

traingle. sinA

2=

s–b s–c

bccos

A

2=

s s–a

bc tan

A

2=

s−b s−c

s s−a

sinB

2=

s–a s–c

accos

B

2=

s s–b

ac tan

B

2=

s−a s−c

s s−b

sinC

2=

s–a s–b

abcos

C

2=

s s–c

ab tan

C

2=

s−a s−b

s s−c

6. Formula that involve the Perimeter: If S=abc

2, where a+b+c is the perimeter of

a traingle, R the radius of the circumcircle, and r the radius of the inscribed circle, then

6. Area of traingle: ∆= s s−a s−b s−c ;(HERO'S FORMULA)

∆=1

2a.b.SinC =

1

2 b.c. sinA =

1

2c.a.sinB=

abc

4R

∆=1

2

a2 sinB. sinC

sinA=

1

2

b2 sin.CsinA

sinB=

1

2

c2 sinA. sinB

sinC =

1

2

a2 sinB.sinC

sin BC

DEDUCTIONS:

sinA=2

bc=

2

bc s s−a s−b s−c sinB=

2

caSinC=

2

ab

tanA

2tan

B

2=

s−c

s; tan

B

2tan

C

2=

s−a

s; tan

C

2tan

A

2=

s−b

s.

tanA

2tan

B

2=

scot

C

2; tan

B

2tan

C

2=

scot

A

2;

tanC

2tan

A

2=

scot

B

2.




OTHER IMPORTANT FORMULA AND CONCEPTS:

1.To find the greatest and least values of the expression asinθ +bcosθ :

Let a=rcosα. b=rsinα , then a2 +b2 =r 2 or r= a2b

2

asinθ +bcosθ = r(sinθ cos α +cosθ sin α) = rsin(θ + α )

But -1≤sin(θ + α )≤1 so that -r ≤rsin(θ + α )≤r. Hence - a2b

2 ≤ asinθ +bcosθ ≤ a2b

2

Thus the greatest and least values of asinθ +bcosθ are respectively a2

b2

and - a2

b2

.

Similarly maximum value of asinθ -bcosθ is a2b2

For 0 , minimum value of a sinθ + bcosecθ is 2 ab

For−

2

2, minimum value of acosθ +bsecθ is 2 ab

For 0

2or

3

2, minimum value of a tanθ +bcotθ is 2

ab

2. cosA.cos2A.cos4A.cos8A............cos2n-1 A =1

2n

sinA

sin 2n A (Remember)

OR cos θ.cos2 θ.cos22 θ.cos23 θ............cos2 n θ =sin 2n1

A

2n sinA

(Each angle being double of preceding)

3. SUM OF THE SIN AND COSINE SERIES WHEN THE ANGLES ARE IN AP:

sinα +sin(α+β) +sin(α +2 β) +..........n terms

cosα +cos(α+β) +cos(α +2 β) +..........n terms

=

sin n.diff

2

sindiff

2

. sin or cos [1st anglelast angle2 ] (Remember the rule)

=

sinn

2

sin

2

.sin or cos [n−1

2 ] =

sinn

2

sin

2

.sin or cos [n−1

2 ]Note: β is not an even multiple of Π i.e. β #2n Π because in that case sum will take the form 0/0. Particular

case: Both the sum will be zero if sinn

2=0 i.e.

n

2=r Π or β =

2r

nor β = even multiple of

nthen S=0

4. SOME RESULTS IN PRODUCT FORM:

sinθ sin(60+θ)sin(60-θ) =1

4sin3θ

cosθ cos(60+θ) cos(60-θ)

=1

4cos3 θ

cosθ cos(120+θ) cos(120-θ)

tanθ tan(60+θ )tan(60-θ ) =tan3θ

sin(600 -A) sin(600 +A) = sin3A

4sinA

cos(600 -A) cos(600 +A)=cos3A

4cosA

tan(600 -A) tan(600 +A) =tan3A

tanA

tan2A tan3A tan5A=tan5A-tan3A-tan2A

tanx tan2x tan3x =tan3x-tan2x-tanx

(Use the above formula at time of integration)

tan(x-α). tan(x+ α ) tan 2x= tan2x-tan(x+ α )-tan(x- α )

4. i) cosA ±sinA= 2sin

4± A = 2cos

4∓ A ii) tanA +cotA =

1

sinA.cosA

5. tan θ + tan

3 + tan

2

3 =3tan3 θ ; tan θ + tan

3 + tan−

3 =3tan3 θ

6. 2 2 2 2............ 22cos2n

=2cos θ ∀n∈ N




HEIGHTS AND DISTANCES-VIGNAN CLASSESANGLE OF ELEVATION AND ANGLE OF DEPRESSIONSuppose a st.line OX is drawn in the horizontal direction.Then the angle XOP where P is a point (or the positionof the object to be observed from the point O of observation )

above OX is called Angle of Elevation of P as seen from O.Similarly, Angle XOQ where Q is below OX, is calledangle of depression of Q as seen from O.

OX is the horizontal line and OP and OQ are called

line of sights

Properties used for solving problems

related to Heights and Distances.1. Any line perpendicular to a plane is

perpendicular to every line lying in the plane.

Explanation: Place your pen PQ upright on your notebook, so that its lower end Q is on the notebook.Through the point Q draw line QA,QB,QC,....... in your notebook in different directions and you willobserve that each of the angles PQA,PQB,..PQC,.... is a right angle. In other words PA is perpendicular to each of the lines QA, QB, QC, lying in the plane.

2.To express one side of a right angled triangle in terms of the other side.

Explanation: Let ABC =Ө, Where ABC is right angledtriangle in which C = 900 . The side opposite to right angle Cwill be denoted by H(Hypotenus),

the side opposite (opposite side) to angle θ is denoted by O,the side containing angle θ (other than H)(Adjacent side) will be denoted by AThen from the figure it is clear thatO=A(tanθ ) or A = O(cotθ ) i.e. Opposite = Adj(tanθ ) or Adj=opposite (cotθ ).Also O=H(sinθ ) or A =H(cosθ ) i.e opposite =Hyp( sinθ ) or Adjacent =Hyp(cosθ )

;,./ []-SWEQRTYUIXCVBNMKL ' 098

PREPARED AND DTP BY KHVASUDEVA,

LECTURER IN MATHEMATICS


O X

Q

α

β

α= Angle of elevation of P

β=Angle of

Depression of Q

H

A

O

θ

THE SPIRIT OF MATHEMATICS

The only way to learn mathematics is to recreate it for oneself -J.L.Kelley

The objects of mathematical study are mental constructs. In order to understand these one

must study , meditate, think and work hard -SHANTHINARYAN

Mathematical theories do not try to find out the true nature of things, that would be anunreasonable aim for them. Their only purpose is to co-ordinate the physical laws we find

from experience but could not even state without the aid of mathematics. -A. POINCARE

Experience and intution, though usually obtained more painfully, may be doveloped by

mathematical insight. -R Aris

Formula- By B B Susheel Kumar

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Transcript of Formula- By B B Susheel Kumar