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) tanAB 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 List Of Formulae By OP Gupta Page - [2] www.theOPGupta.WordPress.com MATHEMATICS – List Of Formulae for Class XII By OP Gupta (+91-9650350480) Inverse Trigonometric Formulae: 1 1 1 1 1 1 01. a)sinx cosec , x 1,1 b) cosecx sin ,x , 1 1, x x 1 1 1 1 1 1 c) cosx sec , x 1,1 d)secx 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)sin1x sin 1 x , x 1,1 b)cos1x π cos 1 x , x 1,1 c) tan1x tan 1 x , x R d) cosec1x cosec 1 x ,|x | 1 e)sec1x π sec 1 x ,|x | 1 f)cot1x π cot 1 x , x R π π 03. a)sin1 sinx x , x b)cos1 cosx x , 0 x π 2 2 π π π π c) tan1 tanx x , x d)cosec1 cosecx x , x , x 0 2 2 2 2 π e)sec1 secx x , 0x π, x f)cot1 cotx x , 0 x π 2 π 04. a)sin1x cos 1 x , x 1,1 2 π b) tan1x cot 1 x , x R 2 π c)cosec1x sec 1 x ,|x | 1 i . e ., x 1 or x 1 2 05. a) sin1x sin 1 y sin 1 x 1 y 2 y 1 x 2 b) cos1x 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 1xy yz zx List Of Formulae By OP Gupta Page - [3] www.theOPGupta.WordPress.com List Of Formulae for Class XII By OP Gupta (Electronics & Communications Engineering) 1 1 2x 06. a) 2 tanx sin2 , | x | 1 1 x 2 1 1 1 x b) 2 tanx cos2 , x 0 1 x 1 1 2x c) 2 tanx tan2 , 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 cos1 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. List Of Formulae By OP Gupta Page - [4] www.theOPGupta.WordPress.com 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 sin1 x is used to denote the smallest angle whether positive or negative, the sine of this angle will give us x. Similarly cos1 x,,,, tan 1 x cosec 1 x sec 1 x and cot1 x are defined. You should note that sin1 x can be written as arcsinx . Similarly other Inverse Trigonometric Functions can also be written as arccosx, arctanx, arcsecx etc. Also note that sin1 x (and similarly other Inverse Trigonometric Functions) is entirely different from ()sin x 1 . In fact, sin1 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.