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ISC Class XII Notes 2024 : Business Studies (Chatrabhuj Narsee Memorial School (CNM), Mumbai)

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Bharti Gupta
Chatrabhuj Narsee Memorial School (CNM), Mumbai

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FOR CLASSES IX, X, XI, & XII CBSE/ICSE/ISC/UP BOARD The pathway to success Trig Cheat Sheet Definition of the Trig Functions Right triangle definition For this definition we assume that 0 < < 2 or 0 < < 90 . Unit circle definition For this definition is any angle. y ( x, y ) hypotenuse opposite y 1 x x adjacent opposite hypotenuse adjacent cos = hypotenuse opposite tan = adjacent sin = hypotenuse opposite hypotenuse sec = adjacent adjacent cot = opposite csc = y =y 1 x cos = = x 1 y tan = x sin = 1 y 1 sec = x x cot = y csc = Facts and Properties Domain The domain is all the values of that can be plugged into the function. sin , can be any angle cos , can be any angle 1 tan , n + , n = 0, 1, 2, 2 csc , n , n = 0, 1, 2, 1 sec , n + , n = 0, 1, 2, 2 cot , n , n = 0, 1, 2, Range The range is all possible values to get out of the function. csc 1 and csc 1 1 sin 1 1 cos 1 sec 1 and sec 1 tan cot Period The period of a function is the number, T, such that f ( + T ) = f ( ) . So, if is a fixed number and is any angle we have the following periods. sin ( ) T= cos ( ) T= tan ( ) T csc ( ) T sec ( ) T cot ( ) T 2 2 = 2 = 2 = = Add:- H . O . 333, SUBHASH NAGAR, BHOLAKHERA ,KRISHNA NAGAR LUCKNOW Ph.# 9169647646: mail id: maaheshwariclasses@gmail.com, website- www.maheshwariclasses.com 1 FOR CLASSES IX, X, XI, & XII CBSE/ICSE/ISC/UP BOARD The pathway to success Formulas and Identities Tangent and Cotangent Identities sin cos tan = cot = cos sin Reciprocal Identities 1 1 csc = sin = sin csc 1 1 sec = cos = cos sec 1 1 cot = tan = tan cot Pythagorean Identities sin 2 + cos 2 = 1 tan + 1 = sec 2 2 1 + cot 2 = csc 2 Even/Odd Formulas sin ( ) = sin csc ( ) = csc cos ( ) = cos sec ( ) = sec tan ( ) = tan cot ( ) = cot Periodic Formulas If n is an integer. sin ( + 2 n ) = sin csc ( + 2 n ) = csc cos ( + 2 n ) = cos sec ( + 2 n ) = sec tan ( + n ) = tan cot ( + n ) = cot Double Angle Formulas sin ( 2 ) = 2sin cos cos ( 2 ) = cos 2 sin 2 = 2 cos 2 1 = 1 2sin 2 tan ( 2 ) = 2 tan 1 tan 2 Degrees to Radians Formulas If x is an angle in degrees and t is an angle in radians then 180t x t = t= and x = 180 x 180 Half Angle Formulas 1 sin 2 = (1 cos ( 2 ) ) 2 1 cos 2 = (1 + cos ( 2 ) ) 2 cos ( 2 ) 1 tan 2 = 1 + cos ( 2 ) Sum and Difference Formulas sin ( ) = sin cos cos sin cos ( ) = cos cos sin sin tan tan 1 tan tan Product to Sum Formulas 1 sin sin = cos ( ) cos ( + ) 2 1 cos cos = cos ( ) + cos ( + ) 2 1 sin cos = sin ( + ) + sin ( ) 2 1 cos sin = sin ( + ) sin ( ) 2 Sum to Product Formulas + sin + sin = 2sin cos 2 2 + sin sin = 2 cos sin 2 2 + cos + cos = 2 cos cos 2 2 + cos cos = 2sin sin 2 2 Cofunction Formulas tan ( ) = sin = cos 2 cos = sin 2 csc = sec 2 sec = csc 2 tan = cot 2 cot = tan 2 Add:- H . O . 333, SUBHASH NAGAR, BHOLAKHERA ,KRISHNA NAGAR LUCKNOW Ph.# 9169647646: mail id: maaheshwariclasses@gmail.com, website- www.maheshwariclasses.com 1 FOR CLASSES IX, X, XI, & XII CBSE/ICSE/ISC/UP BOARD The pathway to success Unit Circle y 1 3 , 2 2 2 2 , 2 2 3 1 , 2 2 3 4 ( 0,1) 2 3 2 90 3 120 2 2 2 , 2 60 4 45 135 5 6 1 3 2 , 2 30 3 1 , 2 2 6 150 ( 1,0 ) 180 3 1 , 2 2 210 7 6 2 2 , 2 2 5 4 0 0 360 2 330 225 4 3 315 7 300 270 4 5 3 3 2 240 1 3 , 2 2 11 6 (1,0 ) x 3 1 , 2 2 2 2 , 2 2 1 3 , 2 2 ( 0, 1) For any ordered pair on the unit circle ( x, y ) : cos = x and sin = y Example 5 cos 3 5 sin 3 1 = 2 3 = 2 Inverse Trig Functions Definition y = sin 1 x is equivalent to x = sin y Inverse Properties cos ( cos 1 ( x ) ) = x cos 1 ( cos ( ) ) = y = cos 1 x is equivalent to x = cos y sin ( sin 1 ( x ) ) = x sin 1 ( sin ( ) ) = tan ( tan 1 ( x ) ) = x tan 1 ( tan ( ) ) = 1 y = tan x is equivalent to x = tan y Domain and Range Function Domain y = sin 1 x 1 1 x 1 y = cos x 1 x 1 y = tan 1 x < x < Range y 2 2 0 y 2 < y< Alternate Notation sin 1 x = arcsin x cos 1 x = arccos x tan 1 x = arctan x 2 Add:- H . O . 333, SUBHASH NAGAR, BHOLAKHERA ,KRISHNA NAGAR LUCKNOW Ph.# 9169647646: mail id: maaheshwariclasses@gmail.com, website- www.maheshwariclasses.com 1 FOR CLASSES IX, X, XI, & XII CBSE/ICSE/ISC/UP BOARD The pathway to success Law of Sines, Cosines and Tangents c a b Law of Sines sin sin sin = = a b c Law of Tangents a b tan 12 ( ) = a + b tan 12 ( + ) Law of Cosines a 2 = b 2 + c 2 2bc cos b c tan 12 ( ) = b + c tan 12 ( + ) b 2 = a 2 + c 2 2ac cos c = a + b 2ab cos 2 2 2 a c tan 12 ( ) = a + c tan 12 ( + ) Mollweide s Formula a + b cos 12 ( ) = c sin 12 Add:- H . O . 333, SUBHASH NAGAR, BHOLAKHERA ,KRISHNA NAGAR LUCKNOW Ph.# 9169647646: mail id: maaheshwariclasses@gmail.com, website- www.maheshwariclasses.com 1

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