Formulae For Trigonometric Functions &

Inverse Trigonometric Functions

 Trigonometric Formulae: Relation between trigonometric ratios sin 1 a) tan  b) tan  c) tan .cot   1 cos cot cos 1 1 d) cot  e) cosec  f) sec  sin sin cos

Trigonometric identities Multiple angle formulae involving 2A and 3A a) sin2 cos 2   1 a)sin2AAA 2sin cos b) 1 tan2  sec 2  AA b) sinA  2sin cos c) 1 cot2  cosec 2  2 2 Addition / subtraction formulae & some related results c) cos2AAA cos2  sin 2

a) sin ABABAB  sin cos  cos sin AA d) cosA  cos2  sin 2 2 2 b) cos ABABAB  cos cos sin sin e) cos2AA 2cos2  1 c) cos ABABABBA cos   cos2  sin 2  cos 2  sin 2 f) 2cos2 AA 1  cos2 d) sin ABABABBA sin   sin2  sin 2  cos 2  cos 2

tanAB tan g) cos2AA 1  2sin2 e) tanAB   1 tanAB tan h) 2sin2 AA 1  cos2 cotBA cot 1 f) cot AB   cotBA cot 2 tan A i)sin 2A  Transformation of sums / differences into products & vice-versa 1 tan2 A CD CD 1 tan2 A a) sinCD sin  2sin cos j) cos2A  2 2 1 tan2 A CD CD b) sinCD sin  2cos sin 2 tan A 2 2 k) tan2A  CD CD 1 tan2 A c)cosCD cos  2cos cos 2 2 CDCD  l)sin3AAA 3sin  4sin3 d) cosCD cos   2sin sin 2 2 e) 2sinAB cos sin ABAB   sin    m) cos3AAA 4cos3  3cos f) 2cosAB sin  sin ABAB   sin   3tanAA tan3 g) 2cosAB cos cos ABAB   cos   n) tan3A  1 3tan2 A h) 2sinAB sin cos ABAB   cos  

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  1o = 60 , 1  = 60 

List Of Formulae By OP Gupta Page - [1] www.theOPGupta.WordPress.com List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering)   1o = = 0.01745 radians approximately   1 radian = 57o 17 45  or 206265 seconds . 180

General Solutions a) sinx sin y  x  n  (  1 )n y , where n  Z . b) cosx cos y  x  2 n  y , where n  Z . c) tanx tan y  x  n  y , where n  Z .

Relation in Degree & Radian Measures Angles in Degree 0 30 45 60 90 180 270 360 c c c c c         c 3  c Angles in Radian 0c             2  6  4  3  2  2 

 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 1 1 3 sin 0 1 2 2 2 cos 3 1 1 1 0 2 2 2 1 tan 0 1 3  3 2 cosec  2 2 1 3 2 sec 1 2 2  3 1 cot  3 1 0 3

Trigonometric Ratios of Allied Angles

Angles    3 3         2  2  T- Ratios  2 2 2 2 OR  sin cos cos sin sin cos  cos sin 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

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

 Inverse Trigonometric Formulae:

1  1 1  1  1 1  01. a)sinx   cosec  , x 1,1  b) cosecx   sin  ,x   ,  1   1,  x  x  1  1 1  1  1 1  c) cosx   sec  , x 1,1  d)secx   cos  ,x   ,  1   1,  x  x 

 1 1   1 1   cot  ,x  0  tan  ,x  0 1  x  1  x  e) tan x    f) cot x     1 1   1 1  π  cot  ,x  0 π  tan  ,x  0  x   x  02. a)sin1x   sin  1 x , x   1,1 b)cos1x π  cos  1 x , x   1,1 c) tan1x   tan  1 x , x  R d) cosec1x   cosec  1 x ,|x | 1 e)sec1x π  sec  1 x ,|x |  1 f)cot1x π  cot  1 x , x  R π π 03. a)sin1  sinx   x ,  x  b)cos1  cosx  x , 0 x  π 2 2 π π π π c) tan1  tanx   x ,  x  d)cosec1  cosecx   x ,  x , x  0 2 2 2 2 π e)sec1  secx   x , 0x  π, x  f)cot1  cotx  x , 0 x  π 2 π 04. a)sin1x cos  1 x  , x   1,1  2 π b) tan1x cot  1 x  , x  R 2 π c)cosec1x sec  1 x  ,|x |  1 i . e ., x   1 or x  1 2

05. a) sin1x sin  1 y  sin  1 x 1  y 2  y 1  x 2   

b) cos1x cos  1 y  cos  1 xy 1  x 2 1  y 2   

 1 x y   tan  ,xy  1  1 xy   1  1  1 x y  c) tanx tan y  π  tan  ,x 0, y  0, xy  1  1 xy   1 x y  π  tan  ,x 0, y  0, xy  1  1 xy 

 1 x y   tan  ,xy   1  1 xy   1  1  1 x y  d) tanx tan y  π  tan  ,x 0, y  0, xy   1  1 xy   1 x y  π  tan  ,x 0, y  0, xy   1  1 xy 

1  1 1  1 x y  z  xyz  e) tanx tan y  tanz  tan   1xy  yz  zx 

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

1  1 2x  06. a) 2 tanx sin2  , | x |  1 1 x  2 1 1 1 x  b) 2 tanx cos2  , x  0 1 x 

1 1 2x  c) 2 tanx tan2  ,  1  x  1 1 x 

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 1 π π  sin x [ 1, 1]  , 2 2  cos1 x [ 1, 1] [0, π]

1 π π  cosec x R (  1, 1) ,  {0} 2 2  1 π  sec x R (  1, 1) [0, π]   2  1 π π  tan x R  ,  2 2  cot 1x R (0, π)

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 3π π  range is the closed interval [–1, 1]. If we restrict its domain to ,  , 2 2  π π  π 3π   , , , etc. then, it becomes bijective with the range [–1, 1]. 2 2  2 2  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 π π  –1 –1 (here except of  , for sin function) are known as the range of sin 2 2  other than its principal value branch. The same discussion can be extended for other inverse circular functions.

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MATHEMATICS – List Of Formulae for Class XII By OP Gupta (+91-9650350480) 09. To simplify inverse trigonometrical expressions, following substitutions can be considered:

Expression Substitution

a2 x 2or a 2  x 2 x atanθ or x  a cot θ

a2 x 2or a 2  x 2 x asin θ or x  a cosθ

x2 a 2or x 2  a 2 x asecθ or x  a cosecθ

a x a  x or x a cos 2θ a x a  x a2 x 2 a 2  x 2 2 2 or x a cos2θ a2 x 2 a 2  x 2 x a x 2 2 or x asin θ or x  a cos θ a x x x a x 2 2 or x atan θ or x  a cot θ a x x

Note the followings and keep them in mind:  The symbol sin1 x is used to denote the smallest angle whether positive or negative, the sine of this angle will give us x. Similarly cos1 x,,,, tan  1 x cosec  1 x sec  1 x and cot1 x are defined.  You should note that sin1 x can be written as arcsinx . Similarly other Inverse Trigonometric Functions can also be written as arccosx, arctanx, arcsecx etc.  Also note that sin1 x (and similarly other Inverse Trigonometric Functions) is entirely different from ()sin x 1 . In fact, sin1 x is the measure of an angle in Radians whose sine is x 1 whereas ()sin x 1 is (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 [email protected]. With lots of Love & Blessings! - OP Gupta [+91-9650 350 480] Electronics & Communications Engineering, Indira Award Winner www.theOPGupta.WordPress.com

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