Formulae For Trigonometric Functions & Inverse ...

[Pages:5]Formulae For Trigonometric Functions & Inverse Trigonometric Functions

Trigonometric Formulae:

Relation between trigonometric ratios

sin a) tan

cos

1 b) tan

cot

cos d) cot

sin

1 e) cosec

sin

c) tan .cot 1

1 f) sec

cos

Trigonometric identities

a) sin2 cos2 1 b) 1 tan2 sec2 c) 1 cot2 cosec2

Addition / subtraction formulae & some related results

Multiple angle formulae involving 2A and 3A

a) sin 2A 2sin Acos A

AA b) sin A 2sin cos

22 c) cos2 A cos2 A sin2 A

a) sin A B sin Acos B cos Asin B b) cos A B cos Acos B sin Asin B c) cos A B cos A B cos2 A sin2 B cos2 B sin2 A d) sin A Bsin A B sin2 A sin2 B cos2 B cos2 A

d) cos A cos2 A sin2 A

2

2

e) cos 2 A 2cos2 A 1

f) 2cos2 A 1 cos2 A

e) tan A B tan A tan B

1 tan A tan B

f) cot A B cot B cot A 1

cot B cot A Transformation of sums / differences into products & vice-versa

g) cos 2 A 1 2sin2 A h) 2sin2 A 1 cos 2A

2 tan A i) sin 2 A

1 tan2 A

a) sinC sin D 2sin C D cos C D

2

2

b) sinC sin D 2cos C D sin C D

2

2

c) cosC cos D 2 cos C D cos C D

2

2

d) cosC cos D 2sin C D sin C D

2

2

e) 2sin Acos B sin A B sin A B

f) 2cos Asin B sin A B sin A B

g) 2cos AcosB cos A B cos A B

h) 2sin Asin B cos A B cos A B

1 tan2 A j) cos 2 A

1 tan2 A 2 tan A

k) tan 2 A 1 tan2 A

l) sin 3A 3sin A 4sin3 A

m) cos 3A 4cos3 A 3cos A 3tan A tan3 A

n) tan 3A 1 3tan2 A

Relations in Different Measures of Angle

Angle in Radian Measure = Angle in Degree Measure?

180 180

Angle in Degree Measure = Angle in Radian Measure?

l (in radian measure) r

Also followings are of importance as well:

1Right angle 90o

1 o = 60, 1 = 60

List Of Formulae By OP Gupta

Page - [1]

theOPGupta.

List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering)

1o = = 0.01745 radians approximately

180

1 radian = 57o1745 or 206265 seconds .

General Solutions

a) sin x sin y x n (1 )n y, where n Z . b) cos x cos y x 2n y, where n Z . c) tan x tan y x n y, where n Z .

Relation in Degree & Radian Measures

Angles in Degree 0 Angles in Radian 0c

30

c

6

45

c

4

60

c

3

90

c

2

180 270 360

c

c

3

2

2 c

In actual practice, we omit the exponent `c' and instead of writing c we simply write and similarly for others.

Trigonometric Ratio of Standard Angles

Degree /Radian

0

30

45

60

90

T ? Ratios

0

6

4

3

2

sin

0

1

1

3

1

2

2

2

cos

1

3

1

1

0

2

2

2

tan

0

1

1

3

3

cosec

2

2 2

1

3

sec

1

2

2

2

3

cot

3

1

1

0

3

Trigonometric Ratios of Allied Angles

Angles

T- Ratios

2

2

sin

cos cos sin

sin

3

2

cos

3

2

cos

2 OR

sin

2

sin

cos

sin sin cos cos sin sin cos cos

tan

cot cot tan tan cot cot tan tan

cot

tan tan cot cot tan tan cot cot

sec

cosec cosec sec sec cosec cosec sec sec

cosec

sec sec cosec cosec sec sec cosec cosec

List Of Formulae By OP Gupta

Page - [2]

theOPGupta.

MATHEMATICS ? List Of Formulae for Class XII By OP Gupta (+91-9650350480)

Inverse Trigonometric Formulae:

01.

a)

sin1

x

cosec1

1

,

x

1,1

x

c)

cos1 x

sec1

1 x

,

x

1,1

b)

cosec1 x

sin1

1

,

x

, 1 1,

x

d) sec1

x

cos1

1 x

,

x

, 1 1,

e)

tan1

x

cot1

1 x

,

x

0

cot1

1 x

,

x

0

f)

cot

1

x

tan1

1 x

,

x

0

tan

1

1 x

,

x

0

02. a) sin1 x sin1 x, x 1,1 c) tan1 x tan1 x, x R e) sec1 x sec1 x, | x | 1

b) cos1 x cos1 x, x 1,1 d) cosec1 x cosec1x, | x | 1 f) cot1 x cot1 x, x R

03.

a) sin1 sin x

x ,

x

2

2

c) tan1 tan x x, x

22

e) sec1 sec x x, 0 x , x

2

b) cos1 cos x x, 0 x

d) cosec1 cosec x x, x , x 0

2

2

f) cot1 cot x x, 0 x

04.

a) sin1 x cos1 x

,

x 1,1

2

b) tan1 x cot1 x

,

x R

2

c) cosec1x sec1 x , | x | 1 i.e., x 1 or x 1 2

05. a) sin1 x sin1 y sin1 x 1 y2 y 1 x2

b) cos1 x cos1 y cos1 xy 1 x2 1 y2

tan1

x y 1 xy

,

xy 1

c) tan1 x tan1 y

tan

1

x y 1 xy

,

x

0,

y 0,

xy 1

tan1

x y 1 xy

,

x 0,

y

0,

xy

1

tan 1

xy 1 xy

,

xy 1

d)

tan

1

x

tan

1

y

tan

1

x y 1 xy

,

x 0,

y 0,

xy 1

tan1

x

y

,

x 0,

y 0,

xy 1

1 xy

e)

tan1

x

tan1

y

tan1

z

tan 1

x y 1 xy

z xyz yz zx

List Of Formulae By OP Gupta

Page - [3]

theOPGupta.

List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering)

06.

a)

2

tan1

x

sin1

1

2x x2

,

|

x

|

1

b) 2 tan1

x

cos1

1 1

x2 x 2

,

x

0

c)

2

tan 1

x

tan1

2x 1 x2

,

1

x

1

07. Principal Value: Numerically smallest angle is known as the principal value.

Finding the principal value: For finding the principal value, following algorithm can be followed?

STEP1? Firstly, draw a trigonometric circle and mark the quadrant in which the angle may lie. STEP2? Select anticlockwise direction for 1st and 2nd quadrants and clockwise direction for 3rd and

4th quadrants. STEP3? Find the angles in the first rotation. STEP4? Select the numerically least (magnitude wise) angle among these two values. The angle

thus found will be the principal value. STEP5? In case, two angles one with positive sign and the other with the negative sign qualify for

the numerically least angle then, it is the convention to select the angle with positive sign as principal value.

The principal value is never numerically greater than .

08. Table demonstrating domains and ranges of Inverse Trigonometric functions:

Inverse Trigonometric Functions i.e., f ( x) Domain/ Values of x Range/ Values of f ( x)

sin1 x

[1, 1]

2

,

2

cos1 x

[1, 1]

[0, ]

cosec1 x sec1 x tan 1 x cot 1x

R (1, 1) R (1, 1)

R R

2

,

2

{0}

[0, ]

2

2

,

2

(0, )

Discussion about the range of inverse circular functions

other than their respective principal value branch

We know that the domain of sine function is the set of real numbers and

range

is

the

closed

interval

[?1,

1].

If

we

restrict

its

domain

to

3 ,

2

2

,

2

,

2

,

3 2 , 2

etc.

then,

it

becomes

bijective

with

the

range

[?1,

1].

So, we can define the inverse of sine function in each of these intervals. Hence, all the intervals of sin?1 function, except principal value branch

(here except

of

2

,

2

for

sin?1

function)

are known as the

range

of

sin?1

other than its principal value branch. The same discussion can be

extended for other inverse circular functions.

List Of Formulae By OP Gupta

Page - [4]

theOPGupta.

MATHEMATICS ? List Of Formulae for Class XII By OP Gupta (+91-9650350480)

09. To simplify inverse trigonometrical expressions, following substitutions can be considered:

Expression

a2 x2 or a2 x2

Substitution x a tan or x a cot

a2 x2 or a2 x2

x a sin or x a cos

x2 a2 or x2 a2

x a sec or x a cosec

ax

ax

or

ax

a x

a2 x2

a2 x2

or

a2 x2

a2 x2

x

ax

or

a x

x

x

ax

or

a x

x

x a cos 2 x2 a2 cos 2 x a sin2 or x a cos2 x a tan2 or x a cot2

Note the followings and keep them in mind: The symbol sin1x is used to denote the smallest angle whether positive or negative, the sine of this angle will give us x. Similarly cos1x, tan1x, cosec1x, sec1x, and cot1x are defined.

You should note that sin1x can be written as arcsinx . Similarly other Inverse Trigonometric Functions can also be written as arccosx, arctanx, arcsecx etc.

Also note that sin1x (and similarly other Inverse Trigonometric Functions) is entirely different from (sin x)1 . In fact, sin1x is the measure of an angle in Radians whose sine is x whereas (sin x)1 is 1 (which is obvious as per the laws of exponents).

sin x

Keep in mind that these inverse trigonometric relations are true only in their domains i.e., they are valid only for some values of `x' for which inverse trigonometric functions are well defined!

Hii, All! I hope this texture may have proved beneficial for you. While going through this material, if you noticed any error(s) or, something which doesn't make sense to you, please bring it in my notice through SMS or Call at +91-9650 350 480 or Email at theopgupta@.

With lots of Love & Blessings!

- OP Gupta [+91-9650 350 480] Electronics & Communications Engineering, Indira Award Winner theOPGupta.

List Of Formulae By OP Gupta

Page - [5]

theOPGupta.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download