Trigonometry Study Material for XI_hsslive_remesh

7/25/2019 Trigonometry Study Material for XI_hsslive_remesh

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STUDY MATERIAL

Prepared by Remesh Chennessery

HSST, HOLYTRINITY EMHSS, TRIVANDRUM.

HSSLIVE.IN

Trigonometry

Angle in Trigonometry

The rotation of a ray from one position to another

along the circumference of a circle is known as an angle.

The initial position of the ray is called initial ray and the

final position of the ray is called terminal ray.

Positve angles: If the rotation of a ray from one position

to another is in anti-clockwise, then the angle is known as

positive angle.

Negative angles: If the rotation of a ray from one

position to another is in anti-clockwise, then the angle is

known as positive angle.

1 Right angle )90(reesdeg90 0

)06(utesmin6010

)06(ondssec601

Radian

One radian is the angle subtended at the centre by a

positive arc equal in length to the radius of the circle.

Thus in the circular system, the circular measure of an

angle = the no. of radians contained in it. It is denoted by

c.

A

O

B

r

r

(=r)

1

c

reedeg180

1radian1 c

= 057 16 22 (approx.)

01deg ree 1 0.01746 radians (approx.)180

Note2: Sexagesimal system. Expression in the form

zyx 0 (x degree y minutes z seconds) is called

sexagesimal system.

Note3:

a) The angle between two consecutive digits in a clock is

030 or

0

12

360

.

b) The hour hand subtend an angle of 030 or

0

12

360

in 1 minute.

c) The minute hand subtend an angle of 06 or

0

60

360

in 1 minute.

Note4: In a regular Polygon

a) All interior angles, exterior angles and all sides are

equal.

b) Sum of all interior angles are equal.

c) Sum of all exterior angles are equal.

d) Each exterior angle =

0

sidesof.No

360

e) Each interior angle = angleExterior180

Note5: To find the angle of a regular polygon means

find its interior angle.


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Degree 30 60 90 120 180 360

Radian6

3

2

3

2 2

Arc, Radius and Angle relation.

Length of the arc of a circle having radius r and

inclination radians is r .

Trigonometric Functions: Trigonometric functions are

periodic and thus many natural phenomena are most

readily studies with the help of trigonometric functions.

Unlike algebraic functions, these functions are not

represented by single letters, instead the abbreviation sin

is used for sine function, cos is for cosine function, tan is for

tangent function, cosec for cosecant function, sec is for

secant function and cot is for cotangent function. For a

given angle , it is usual to write sinfor sin(), etc.

sin=r

y

hypotenuse

side.opp

cos=r

x

hypotenuse

sideadjacent

tan= x

y

sideadjacent

side.opp

cosec=y

r

sin

1

sec=x

r

cos

1

cot=y

x

tan

1

Quotient Relations

tan

cos

sin

cotsin

cos

Phythagoras relations

i. 1sincos 22

a) 22 sin1cos

b) 22 cos1sin

ii. 1tansec 22

a) 22 tan1sec

b) 1sectan 22

iii. 1coteccos 22

a) 22 cot1eccos

b) 1eccoscot 22

Quadrants

Two mutually perpendicular lines in a plane divide the

plane into 4 regions and each region is called quadrant.

X

Y

O

P

M

y

x

r

r

rO

A

B

l


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Signs of Trigonometric functions in different

Quadrants

Hint for memory:-

1. All Students Take Coffee

I II III IV

2. After School Tuition Class

Note:

Acute angle - angle less than 900[< 900]

Obtuse angle - angle greater than 900[> 900]

Right angle - angle is equal to 900[ = 900]

Trigonometric Ratios of particular angles.

090cos0sin 190sin0cos

090cot0tan 90tan0cot

2

160cos30sin

2

360sin30cos

3

160cot30tan 360tan30cot

2

145cos45sin 145cot45tan

Compound Angles and Reduction Formulae

BsinAcosBcosAsinBAsin

BsinAcosBcosAsinBAsin

BsinAsinBcosAcosBAcos

BsinAsinBcosAcosBAcos [Proof needed]

BtanAtan1

BtanAtanBAtan

BtanAtan1

BtanAtanBAtan

BcotAcot

1BcotAcotBAcot

AcotBcot

1BcotAcotBAcot

Atan1

Atan1)A45tan(

Atan1

Atan1)A45tan(

CsinBsinAsinCcosBcosAsinCBAsin

CBACBACBA cossinsincoscoscoscos

BtanAtan1

CtanBtanAtanAtanCBAtan

Important Formulae

BsinAsinBAsinBAsin 22 [Proof needed]

BsinAcosBAcos)BAcos( 22 [ -do- ]

BtanAtan1

BtanAtanBAtan)BAtan(

22

22

Product Formulae.

i.

2

DCcos2

DCsin2DsinCsin

ii.

2

DCsin

2

DCcos2DsinCsin

iii.

2

DCcos

2

DCcos2DcosCcos

iv.

2

DCsin

2

DCsin2DcosCcos

XX

Y

Y

O

I quadrant

IV quadrant

II quadrant

III quadrant

All+vesin, cosecare +ve

tan, cotare +ve

cos, secare +ve

XX

Y

Y

O

I quadrant

IV quadrant

II quadrant

III quadrant

90 360 +

90 + 180

180 + 270 270 +

360


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Converse of Product Formulae.

i. BAsinBAsinBcosAsin2

ii. BAsinBAsinBsinAcos2

iii. BAcosBAcosBcosAcos2

iv. BAcosBAcosBsinAsin2

Related angles & General Reduction Formulae

For any angle , the angles ,180,90,

360,270 , etc. are called related angles and the

functions expressed in terms of the related angles are

known as General Reduction Formulae.

Reduction Formulae for (

)

sinsin : eccoseccos

coscos : secsec

tantan : cotcot

Reduction Formulae for (90

)

cos90sin : sin90cos

cot90tan : tan90cot

Reduction Formulae for (90+)

sin 90 cos : cos 90 sin

tan 90 cot : cot 90 tan

Reduction Formulae for (180

)

sin180sin : cos180cos

tan180tan : cot180cot

sin180sin : cos180cos

tan180tan : cot180cot

Hint: Any trigonometric function is in the form 90.n is

numerically equal to:

1) the same function if n is an even integer (2,4,6,)

2) the corresponding co-function if n is an odd

integer (1,3,5, )

The algebraic sign in each case is same as the

quadrant in which the given angle 90.n lies.

Function Co-function

sin cos

cos sin

tan cot

cot tan

eccos sec sec eccos

Examples:

2

160cos)6090.1sin(150sin (or)

2

130sin)3090.2sin(150sin

2

145sin)4590.2sin(225sin

2330cos)3090.4cos(330cos

Note: Since T-functions are circular functions, we can add

or subtract any number of 360s to the given function. Its

value is not changed. For example

2

360sin)360420sin(420sin

)720765sin()360.2765sin()765sin(

=2

145sin)45sin(

Multiple Angles

Atan1

Atan2AcosAsin2A2sin

2

1Acos2Asin21AsinAcosA2cos 2222

=Atan1

Atan12

2

Atan1

Atan2A2tan

2

Acos2A2cos1

2

: Asin2A2cos1 2

2AsinAcosA2sin1 : 2AsinAcosA2sin1

AtanA2cos1

A2cos1 2

:Atan

1Atan1Atan

2

4

1518sin

:

4

1536cos


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4

521018cos

:

4

521036sin

AcotA2cos1

A2cos1 2

:

A4

tanA2sin1

A2sin1 2

A4

tanA2sin1

A2sin1 2

Asin4Asin3A3sin 3

Acos3Acos4A3cos 3

Atan31

AtanAtan3A3tan

2

3

n2 n

n

s in2 AcosA cos2Acos2 A...cos2 A

2 sinA

Sub-Multiple Angles

2

Atan1

2

Atan2

2

Acos

2

Asin2Asin

2

12

Acos2

2

Asin21

2

Asin

2

AcosAcos

2222

=

2

Atan1

2

Atan1

2

2

2

Atan1

2

Atan2

Atan

2

2

Acos2Acos1

2

2

Asin2Acos1

2

2

2

Asin

2

AcosAsin1

2

2

Asin

2

AcosAsin1

2

Atan

Acos1

Acos1 2

2

Acot

Acos1

Acos1 2

2

A

4tan

Asin1

Asin1 2

2

A

4tan

Asin1

Asin1 2

1 cos AsinA Atan 2

21 tan A 1

tanA

1 cos A

sinA

Acot

2

21 tan A 1

tanA

Values of0 0 0 01 1 1 1

sin 22 , cos 22 , tan 22 , cot 222 2 2 2

01 2 2

sin222 2

01 2 2

cos222 2

01

tan 22 2 12

01

cot 22 2 12

Values of sin90, cos90, tan90

0 3 5 5 5sin9

4

0 3 5 5 5cos9

4

0 3 5 5 5tan9

3 5 5 5

Values of sin150, cos150, tan150

0 3 1sin152 2

0 3 1cos15

2 2

0tan15 2 3

Values of sin750, cos750, tan750, cot750

3 1sin75

2 2

3 1cos75

2 2

3 1tan75

3 1

3 1cot 75

3 1

1sin sin sin sin 3

3 3 4


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1cos cos cos cos3

3 3 4

tan tan tan tan 33 3

1cos36 cos72

2 :

1cos36cos72

4

Trigonometric Equations.

The general value of if

i. sin=0 is = n, nZ

ii. cos=0 is = 2

.1n2

, nZ

iii. tan=0 is = n, nZ


i. sin= sinis = n+ (1)n., nZ

ii. cos= cosis = 2n , nZ

iii. tan= tanis = n+ , nZ


i. 22 sinsin is = n

ii. 22 coscos is = n

iii. 22 tantan is = n

The general value of if csinbcosa

dividing throughout by22

ba then

222222ba

csin

ba

bcos

ba

a

. (1)

knowing the values of a and b we can find the value of

the angle such that22

ba

acos

and

22ba

bsin

. Also find the angle such that

22ba

ccos

. Then (1) becomes,

cossinsincoscos i.e., cossinsincoscos

i.e., coscos

n2

n2

Trigonometric Identities

i. In any ABC, A+B+C =

Then A + B = C

C = (A+B)

ii. In any ABC, A+B+C =

Then2

BA=

2 2

C

2

C=

2 2

A B

Tips1:

1. ......,3,2,,0then,0sin

2. ......,3,2,,0then,0tan

3. ......,4,2,0then,1cos

4. .........,2

5,

2

3,

2then,0cos

5. .......,5,3,then,1cos

Tips2:

=

Then =

=

Then = +

=

Then = 2

=

Then = + =

Then = +

=

Then = 2

Domain and Range of trigonometric functions:

f(x) Domain Range

sin x R 1, 1

cos x R 1, 1

tanx R 2n 1 , n Z2

R

cosecx R n , n Z , 1 1, secx R 2n 1 , n Z

2

, 1 1,

cotx R n , n Z R


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Graph of t-functions

1. f ( x) sin x

47

2

3

5

2

2

3

2

2

0

2

3

2

2 5

2

3 7

2

4

0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0

2. f ( x) cos x

47

2

35

2

23

2

2

0

2

3

2

2 5

2

3 7

2

4

1 0 -1 0 1 0 -1 0 1 0 -1 0 1 0 -1 0 1

4 7

2

3 5

2

3

2

2

O

472

35

2

23

2

2

X

2 X

Y

Y

3. f ( x) tan x

2

0

2

3

2

2

-1 1 0 0

4 7

2

3

5

2

2 3

2

2

O

472

35

2

23

2

2

XX

Y

Y

-1

1


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XO

2

32

22

X

Y

Y

4. f ( x) cot x

2

0

2

3

2

2

0 0 0

2

2

O 3

2

2

X

Y

X

Y

5. f ( x) secx

2

0

2

3

2

2

1 1 1 1


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2

O2

3

2

2

6. f(x) cosecx

2

0

2

3

2

2

1 1 1

O 3

2

22

2


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Properties of les

Properti es of tri angles are not i ncluded in the cur rent syll abus of class XI and XI I . Bu t the awareness of this topic

will help you to wr ite all types of competiti ve examinations.

i. Sine rule: In any ABC,Csin

c

Bsin

b

Asin

a , where a, b, c are the sides and A, B, C are the angles.

ii. Cosine rule: In any ABC,

a2= b2+ c2 2bc cosA

b2= c2+ a2 2ca cosB

c2= a2+ b2 2ab cosC

iii. Tangent rule(Napiers Formula)

In any ABC,

2

Acot

cb

cb

2

CBtan

2

Bcot

ac

ac

2

ACtan

2

Ccot

ba

ba

2

BAtan

Projection Formulae

In any ABC,

BcoscCcosba

CcosaAcoscb

AcosbBcosac

Heros Formula

In any ABC,

Area = s s a s b s c , where,a b c

s2

The sine of the angle in terms of the sides

i. Asin =bc

2 csbsass

ii. Bsin =ca

2 csbsass

iii. Csin =ab

2 csbsass


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T-Functions of half angles in terms of the sides.

2

Asin =

bc

csbs :

2

Bsin =

ca

ascs

2

Csin =

ab

bsas :

2

Acos =

bc

ass

2

Bcos =

ca

bss :

2

Ccos =

ab

css

2

Atan =

ass

csbs

:

2

Btan =

bss

ascs

2

Ctan =

css

bsas

Area of a Triangle

i. Asinbc2

1 (or) Bsinca

2

1 (or) Csinab

2

1

ii. rs

The other forms of

abc

4R

22R sin Asin BsinC 21 sin B sin C

a

2 sin A

21 sin C sin Ab2 sin B

21 sin A sin B

c2 sin C

The radius of the incircle of a triangle (Half Angle Formulae)

The radius of the incircle, 2

Atanasr (or)

2

Btanbsr (or)

2

Ctancsr

Circum-radius of a Triangle

In any ABC,Csin

c

Bsin

b

Asin

a where R is the circumradius of the circum circle.

a 2R sin A b 2R sin B c 2R sin C

Other form of R:abc

R4

Note:


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a b c 2s : a b c 2a 2s 2a

a b c 2b 2s 2b : a b c 2c 2s 2c

Circum radius of a circle.

Asin2

aR =

Bsin2

b=

Csin2

c

Note:

In any ABC,

1. AsinR2a : BsinR2b : c= CsinR2

2.bc2

acba

2222 :

ca2

bacb

2222

ab2

cba

c

2222

Some Important relations: In any ABC,

abc 4Rrs :1 1 1 1

ab bc ca 2R r

abca b c

2R r :

ssinA sinB sinC

R 4R

rcos A cos B cos C 1

R : 2sinAsinBsinC 2R

2 2 2 2

a b c 8R : inradius of a ABC , r s

A

r s a tan2

: B

r s b tan2

C

r s c tan2

:

B Ca sin sin

2 2rA

cos2

C Ab sin sin

2 2rB

cos2

:

A Bc sin sin

2 2rC

cos2

A B Cr 4R sin sin sin

2 2 2 : 2

A B Ccot cot cot

2 2 2 r

1 1 1 4R

s a s b s c

:A B C

4Rr cos cos cos2 2 2

A B C4R cos cos cos s

2 2 2

Tabular Logarithm


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The tabular logarithm of a trigonometric ratio is obtained by adding 10 to the logarithm of that ratio.

L sin log sin 10

L cos log cos 10

L tan log tan 10

E.g.: Lsin35 43 22 logsin35 43 22 10 1.7663 10 9.7663

More Tips in Trigonometry

1. 1 2 3 n 1 2 3 n 1 3 5 7sin A A A ....... A cos A cos A cosA ... cos A S S S S ...

2. 1 2 3 n 1 2 3 n 2 4 6cos A A A ....... A cos A cos A cos A ... cos A 1 S S S ...

3. 1 3 5 71 2 3 n2 4 6

S S S S ...tan A A A ....... A

1 S S S ...

where

1 1 2 nS Sum of the tangent of the separate angles tanA tanA ... tanA

2 1 2 2 3S Sum of the tangents taken two at a time tanA tanA tanA tanA ...

3 1 2 3 2 3 4S Sum of the tangents taken three at a time tanA tanA tanA tanA tanA tanA ...

Note: If n 2 n 31 2 3 n 1 2 2 3 3A A A ... A A, then S n tan A, S C tan A, S C tan A,... . Then

4. n n n 3 n 51 3 5sin nA cos A C tan A C tan A C tan A ... 5. ...tantan1coscos 4422 ACACAnA nnn

6.

.. .tantantan1

.. .tantantantan

6

6

4

4

2

2

55

331

ACACAC

ACACACnA

nnn

nnn

7. ...tantantantantan1coscossin 554433221 ACACACACACAnAnA nnnnnn 8. ...tantantantantantan1coscossin 66554433221 ACACACACACACAnAnA nnnnnnn

9.

2

sin.

2sin

21sin

1sin.. .2sinsinsin n

n

n

10.

2

sin.

2

sin

21cos

1cos.. .2coscoscos n

n

n

Chances favour only the mind that are prepared Louis Pasteur.

Trigonometry Study Material for XI_hsslive_remesh

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