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Class 12 Exam 2017 : English Paper 1 (English Language)

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Raju Raj
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Trigonometric Formula Sheet Definition of the Trig Functions Unit Circle Definition Assume can be any angle. Right Triangle Definition Assume that: or 0 < < 90 0< < 2 y (x, y) 1 hypotenuse opposite y x x adjacent opp hyp adj cos = hyp opp tan = adj sin = hyp opp hyp sec = adj adj cot = opp y 1 x cos = 1 y tan = x csc = sin = 1 y 1 sec = x x cot = y csc = Domains of the Trig Functions sin , ( , ) csc , = n , where n Z cos , ( , ) sec , = n+ tan , = n+ cot , = n , where n Z 1 , where n Z 2 1 , where n Z 2 Ranges of the Trig Functions 1 sin 1 1 cos 1 tan csc 1 and csc 1 sec 1 and sec 1 cot Periods of the Trig Functions 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. 2 2 sec( ) T = cot( ) T = 2 2 cos( ) T = tan( ) T = csc( ) T = sin( ) T = 1 Identities and Formulas Tangent and Cotangent Identities tan = sin cos cot = Half Angle Formulas cos sin sin = cos = 1 csc 1 cos = sec 1 tan = cot sin = 1 sin 1 sec = cos 1 cot = tan csc = sin( ) = sin cos cos sin cos( ) = cos cos sin sin 2 sin + cos = 1 tan2 + 1 = sec2 tan( ) = 1 + cot2 = csc2 1 sin sin = [cos( ) cos( + )] 2 1 cos cos = [cos( ) + cos( + )] 2 1 sin cos = [sin( + ) + sin( )] 2 1 cos sin = [sin( + ) sin( )] 2 csc( ) = csc sec( ) = sec cot( ) = cot Periodic Formulas If n is an integer sin( + 2 n) = sin cos( + 2 n) = cos tan( + n) = tan csc( + 2 n) = csc sec( + 2 n) = sec cot( + n) = cot Sum to Product Formulas + cos 2 2 + sin sin = 2 cos sin 2 2 + cos + cos = 2 cos cos 2 2 + cos cos = 2 sin sin 2 2 sin + sin = 2 sin Double Angle Formulas sin(2 ) = 2 sin cos cos(2 ) = cos2 sin2 = 2 cos2 1 = 1 2 sin2 tan(2 ) = tan tan 1 tan tan Product to Sum Formulas Even and Odd Formulas sin( ) = sin cos( ) = cos tan( ) = tan 1 cos(2 ) 1 + cos(2 ) Sum and Difference Formulas Pythagorean Identities 2 1 + cos(2 ) 2 tan = Reciprocal Identities 1 cos(2 ) 2 2 tan 1 tan2 Cofunction Formulas Degrees to Radians Formulas sin = cos 2 If x is an angle in degrees and t is an angle in radians then: csc = sec 2 t x 180 t = t = and x = tan = cot 180 x 180 2 2 = sin 2 sec = csc 2 cot = tan 2 cos Unit Circle (0, 1) , ( 1 2 ( 3 ) 2 , (1 2 3 ) 2 2 2 , ) 2 2 ( 60 , 3 120 , 2 3 2 2 , ) 2 2 ( 3 1 , ) 2 2 30 , 6 150 , 5 6 ( 1, 0) 45 , 4 135 , 3 4 3 1 , ) 2 2 ( 90 , 2 180 , 0 , 2 210 , 7 6 ( 3 , 1 ) 2 2 ( 330 , 11 6 225 , 5 4 240 , 4 3 2 , 22 ) 2 300 , 5 3 ( ( 1 , 2 3 ) 2 270 , 3 2 (1 , 2 (0, 1) F or any ordered pair on the unit circle (x, y) : cos = x and sin = y Example cos ( 7 6) ( 315 , 7 4 = 23 1 sin ( 7 6 ) = 2 3 3 ) 2 3 1 , 2 ) 2 2 2 , ) 2 2 (1, 0) Inverse Trig Functions Definition Inverse Properties These properties hold for x in the domain and in the range = sin 1 (x) is equivalent to x = sin = cos 1 (x) is equivalent to x = cos sin(sin 1 (x)) = x cos(cos 1 (x)) = x Domain and Range Function Domain 1 x 1 tan 1 (tan( )) = Range = sin 1 (x) cos 1 (cos( )) = tan(tan 1 (x)) = x = tan 1 (x) is equivalent to x = tan sin 1 (sin( )) = 1 = cos (x) 1 = tan (x) Other Notations 2 2 sin 1 (x) = arcsin(x) 0 1 x 1 x cos 1 (x) = arccos(x) < < 2 2 tan 1 (x) = arctan(x) Law of Sines, Cosines, and Tangents a c b Law of Sines Law of Tangents tan 1 ( ) a b 2 = 1 a+b tan 2 ( + ) sin sin sin = = a b c Law of Cosines tan 1 ( ) b c 2 = 1 b+c tan 2 ( + ) a2 = b2 + c2 2bc cos b2 = a2 + c2 2ac cos tan 1 ( ) a c 2 = 1 a+c tan 2 ( + ) c2 = a2 + b2 2ab cos 4 Complex Numbers i= a = i a, a 0 i2 = 1 1 i3 = i i4 = 1 (a + bi)(a bi) = a2 + b2 a2 + b2 Complex Modulus (a + bi) + (c + di) = a + c + (b + d)i |a + bi| = (a + bi) (c + di) = a c + (b d)i (a + bi) = a bi Complex Conjugate (a + bi)(c + di) = ac bd + (ad + bc)i (a + bi)(a + bi) = |a + bi|2 DeMoivre s Theorem Let z = r(cos + i sin ), and let n be a positive integer. Then: z n = rn (cos n + i sin n ). Example: Let z = 1 i, find z 6 . Solution: First write z in polar form. r= (1)2 + ( 1)2 = = arg(z) = tan 1 Polar Form: z = 2 1 1 = 4 + i sin 2 cos 4 4 Applying DeMoivre s Theorem gives : z6 = 2 6 + i sin 6 4 4 3 + i sin 2 cos 6 = 23 cos 3 2 = 8(0 + i(1)) = 8i 5 Finding the nth roots of a number using DeMoivre s Theorem Example: Find all the complex fourth roots of 4. That is, find all the complex solutions of x4 = 4. We are asked to find all complex fourth roots of 4. These are all the solutions (including the complex values) of the equation x4 = 4. For any positive integer n , a nonzero complex number z has exactly n distinct nth roots. More specifically, if z is written in the trigonometric form r(cos + i sin ), the nth roots of z are given by the following formula. ( ) r 1 n 360 k + cos n n 360 k + + i sin n n , f or k = 0, 1, 2, ..., n 1. Remember from the previous example we need to write 4 in trigonometric form by using: b r = (a)2 + (b)2 . and = arg(z) = tan 1 a So we have the complex number a + ib = 4 + i0. Therefore a = 4 and b = 0 (4)2 + (0)2 = 4 and 0 = arg(z) = tan 1 =0 4 Finally our trigonometric form is 4 = 4(cos 0 + i sin 0 ) So r = Using the formula ( ) above with n = 4, we can find the fourth roots of 4(cos 0 + i sin 0 ) 1 For k = 0, 4 4 1 For k = 1, 4 4 1 For k = 2, 4 4 1 For k = 3, 4 4 cos cos cos cos 0 360 0 + + i sin 4 4 0 360 1 + + i sin 4 4 0 360 2 + + i sin 4 4 0 360 3 + + i sin 4 4 0 360 0 + 4 4 0 360 1 + 4 4 0 360 2 + 4 4 0 360 3 + 4 4 Thus all of the complex roots of x4 = 4 are: 2, 2i, 2, 2i . 6 = = 2 (cos(90 ) + i sin(90 )) = 2i = 2 (cos(180 ) + i sin(180 )) = 2 = 2 (cos(270 ) + i sin(270 )) = 2i 2 (cos(0 ) + i sin(0 )) = 2 Formulas for the Conic Sections Circle StandardF orm : (x h)2 + (y k)2 = r2 W here (h, k) = center and r = radius Ellipse Standard F orm f or Horizontal M ajor Axis : (x h)2 (y k)2 + =1 a2 b2 Standard F orm f or V ertical M ajor Axis : (x h)2 (y k)2 + =1 b2 a2 Where (h, k)= center 2a=length of major axis 2b=length of minor axis (0 < b < a) Foci can be found by using c2 = a2 b2 Where c= foci length 7 More Conic Sections Hyperbola Standard F orm f or Horizontal T ransverse Axis : (x h)2 (y k)2 =1 a2 b2 Standard F orm f or V ertical T ransverse Axis : (y k)2 (x h)2 =1 a2 b2 Where (h, k)= center a=distance between center and either vertex Foci can be found by using b2 = c2 a2 Where c is the distance between center and either focus. (b > 0) Parabola Vertical axis: y = a(x h)2 + k Horizontal axis: x = a(y k)2 + h Where (h, k)= vertex a=scaling factor 8 f (x) f (x) = sin(x) 1 3 2 2 2 1 2 x 0 6 4 3 2 2 3 3 4 5 6 7 6 5 4 4 3 3 2 5 3 7 4 11 6 2 4 3 3 2 5 3 7 4 11 6 2 1 2 2 2 23 -1 5 2 = Example : sin 4 2 f (x) f (x) = cos(x) 1 3 2 2 2 1 2 x 0 6 4 3 2 2 3 3 4 5 6 7 6 5 4 1 2 2 2 23 -1 7 3 = Example : cos 6 2 9 2 2 f (x) f (x) = tan x 3 1 3 3 3 2 5 6 4 3 3 4 6 0 3 3 1 3 10 x 6 4 3 2 3 3 4 5 6

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