Theory Class XI TARGET : JEE Main/Adv PREPARED BY: ER. VINEET LOOMBA (B.TECH. IIT ROORKEE) MATHEMATICS Trigonometry SHARING IS CARING!! Want to Thank me? Share this Assignment with your friends and show them that you care for them. Enjoy!! Trigonometrical Ratios & Identities 1. Basic Trigonometric Identities: (A) sin² + cos² = 1; 1 sin 1; 1 cos 1 R (B) sec² tan² = 1 ; sec 1 R n 1, n (C) cosec² cot² = 1 ; cosec 1 R n, n. Circular Definition Of Trigonometric Functions: PM sin = OP OM cos = OP sin tan =, cos 0 cos cos cot =, sin 0 sin 1 1 sec =, cos 0cosec =, sin 0 cos sin 3. Trigonometric Functions Of Allied Angles: If is any angle, then 90 ±, 180 ±, 70 ±, 360 ± etc. are called ALLIED ANGLES. (A) sin ( ) = sin ; cos ( ) = cos (B) sin (90 ) = cos ;cos (90 ) = sin (C) sin (90 + ) = cos ;cos (90 + ) = sin (D) sin (180 ) = sin ;cos (180 ) = cos (e) sin (180 + ) = sin ;cos (180 + ) = cos
Class (XI) (f) sin (70 ) = cos (g) sin (70 + ) = cos (h) tan (90 ) = cot ;cos (70 ) = sin ;cos (70 + ) = sin ;cot (90 ) = tan 4. Graphs of Trigonometric functions: (A) y = sin x x R; y [ 1, 1] (B) y = cos x x R; y [ 1, 1] (C) y = tan x x R (n + 1)/, n ; y R (D) y = cot x x R n, n ; y R (e) y = cosec xx R n, n ; y (, 1] [1, )
Jupiter (XI) 3 (f) y = sec x x R (n + 1)/, n ; y (, 1] [1, ) 5. Trigonometric Functions of Sum or Difference of Two Angles: (A) sin (A ± B) = sina cosb ± cosa sinb (B) cos (A ± B) = cosa cosb sina sinb (C) sin²a sin²b = cos²b cos²a = sin (A+B). sin (A B) (D) cos²a sin²b = cos²b sin²a = cos (A+B). cos (A B) (e) tan (A ± B) = tan A tanb 1 tan A tanb cot A cot B 1 (f) cot (A ± B) = cot B cot A (g) tan (A + B + C) tan A tanb tanctan A tanb tanc =. 1 tana tanb tanb tanc tanc tan A 6. Factorisation of the Sum or Difference of Two Sines or Cosines: CD CD (A) sinc + sind = sin cos (B) (C) CD CD sinc sind = cos sin CD CD cosc + cosd = cos cos (D) cosc cosd = CD CD sin sin 7. Transformation of Products into Sum or Difference of Sines & Cosines: (A) sina cosb = sin(a+b) + sin(ab) (B) cosa sinb = sin(a+b) sin(ab) (C) cosa cosb = cos(a+b) + cos(ab) (D) sina sinb = cos(ab) cos(a+b) 8. Multiple and Sub-multiple Angles : (A) sin A = sina cosa ; sin = sin cos (B) cos A = cos²a sin²a = cos²a 1 = 1 sin²a;
Jupiter (XI) 4 cos² = 1 + cos, sin² = 1 cos. tan A (C) tan A = 1 tan A tan A (D) sin A = 1 tan A tan ; tan = 1 tan 1 tan A, cos A = 1 tan A (e) (f) sin 3A = 3 sina 4 sin 3 A cos 3A = 4 cos 3 A 3 cosa (g) tan 3A = 3 tan A tan 1 3 tan 9. Important Trigonometric Ratios: A 3 A (A) sin n = 0 ; cos n = (1) n ; tan n = 0, where n (B) sin 15 or sin 1 cos 15 or cos 1 tan 15 = 51 (C) sin or sin 18 = 10 4 10. Conditional Identities: If A + B + C = then : = = 3 1 5 = cos 75 or cos 1 3 1 5 = sin 75 or sin 1 3 1 = 3 = cot 75 ; tan 75 = 31 (i) sina + sinb + sinc = 4 sina sinb sinc (ii) sina + sinb + sinc = 4 cos A cos B cos C (iii) (iv) (v) 51 & cos 36 or cos = 5 4 cos A + cos B + cos C = 1 4 cos A cos B cos C cos A + cos B + cos C = 1 + 4 sin A sin B sin C tana + tanb + tanc = tana tanb tanc (vi) tan A tan B + tan B tan C + tan C tan A = 1 (vii) cot A + cot B + cot C = cot A. cot B. cot C (viii) cot A cot B + cot B cot C + cot C cot A = 1 ; ; 3 1 = 3 = cot 15 31 (ix) A + B + C = then tan A tan B + tan B tan C + tan C tan A = 1
Jupiter (XI) 5 11. Range of Trigonometric Expression: E = a sin + b cos E = a b sin ( + ), where tan = a b = a a b cos (), where tan = b Hence for any real value of, a b E a b 1. Sine and Cosine Series: n sin sin + sin (+) + sin ( + ) +... + sin n 1 = sin n sin cos + cos (+) + cos ( + ) +... + cos n 1 = sin 1. DEFINITION n 1 sin n 1 cos The equations involving trigonometric function of unknown angles are known as Trigonometric equations e.g. cos = 0, cos 4cos =1, sin + sin =, cos 4sin =1 A solution of a trigonometric equation is the value of the unknown angle that satisfies the equation. e.g., 1 3 9 11 sin or,,,... 4 4 4 4 4. PERIODIC FUNCTION A function f(x) is said to be periodic if there exists T > 0 such that f(x + T) = f(x) for all x in the domain of definitions of f(x). If T is the smallest positive real numbers such that f(x + T) = f(x), then it is called the period of f(x) Since sin (n + x ) = sinx, cos (n + x) = cos x ; tan (n + x) = tan x for all n Z Therefore sinx, cosx and tanx are perodic function, the period of sinx and cos x is and that of tanx is. Function sin (ax + b), cos (ax +b), sec (ax + b), cosec (ax +b ) tan (ax + b), cot (ax +b) Trigonometrical Equations sin (ax + b), cos (ax +b), sec (ax +b), cosec (ax +b ) /a Period /a tan (ax + b ), cot (ax +b ) /a /a
Jupiter (XI) 6 3. TRIGONOMETRICAL EQUATIONS WITH THEIR GENERAL SOLUTION Trigonometrical equation If sin = 0 then = n General solution If cos = 0 then = (n + / ) = (n+1)/ If tan = 0 then = n If sin = 1 then = n + / = (4n+1)/ If cos = 1 then = n If sin = sin then = n + ( 1) n where [ /, /] If cos = cos then = n where [0, ] If tan = tan then = n + where ( /, /) If sin = sin then = n If cos = cos then = n If tan = tan then = n If sin sin * cos cos If sin sin * tan tan If tan tan * cos cos then then then = n + = n + = n + * Every where in this chapter "n" is taken as an integer. * If be the least positive value of which statisfy two given trigonometrical equations, then the general value of will be n + 4. GENERAL SOLUTION OF STANDARD TRIGONOMETRICAL EQUATIONS Since, trigonometric functions are periodic, The solution consisting of all possible solutions of a trigonometric equation is called its general solution. We use of following results for solving the trigonometric equations ; Result 1 : sin = 0 = n, n I. We know that sin = 0 for all integral multiples of. (by graphical approach) sin = 0 = 0,,, 3,... = n, n I. sin = 0 = n, n I.
Jupiter (XI) 7 Result : cos = 0 = (n +1), n I. We know that cos = 0 for all odd multiples of (by graphical approch) cos = 0 = 3, 5,... = (n +1), n I. cos = 0 = (n+1), n I. Result 3 : tan = 0 = n, n I. We know that tan = 0 for all integral multiple of. tan = 0 = 0,,, 3,... = n, n I. tan = 0 = n, n I. Result 4 : sin = sin = n + ( 1) n, where n I and,. We have, sin sin = 0 sin = sin, where, cos sin = 0 cos = 0 or sin = 0 = (m +1), m I or = m, m I. ( + ) = (m + 1), m I or ( ) = m, m I = (m + 1), m I or = (m ) +, m I = ( any odd multiple of ) or = (any even multiple of ) + =n + ( 1) n, where n I sin = sin = n + ( 1) n, where ni and,
Jupiter (XI) 8 Result 5 : cos = cos = n, n I and [ 0, ] We have, cos = cos, where 0, cos cos = 0 sin. sin = 0 sin = 0 or sin = 0 = n or = n, n I. + = n or = n, n I = n or = n +, n I cos = cos = n, n I = n, n I, where [ 0, ] Result 6 : tan = tan = n +, n I where, we have tan = tan, where, sin sin cos cos sin cos cos sin = 0 sin ( ) = 0 = n, n I = n +, n I tan = tan = n + where 5. GENERAL SOLUTION OF THE TRIGONOMETRICAL EQUATION, Result 7 : sin = sin, cos = cos, tan = tan = n (i) sin = sin 1cos 1cos cos = cos = n, n I = n, n I
Jupiter (XI) 9 (ii) cos = cos (iii) tan = tan 1cos 1cos cos = cos = n, n I = n, n I 1 tan 1 tan 1 tan 1 tan cos = cos = n \, n I = n, n I (applying componendo and dividendo) 6. GENERAL SOLUTION OF TRIGONOMETRICAL EQUATION a cos + b sin = c To solve the equation a cos + b sin = c, put a = r cos, b = r sin such that r a b, tan a 1 b Substituting these values in the equation we have r cos cos r sin sin = c cos If c > a b, then the equation ; c r a cos + b sin = c has no solution If c a b, then put ; c a b = cos, so that cos ( ) = cos n n cos 7. SOLUTIONS IN THE CASE OF TWO EQUATIONS ARE GIVEN Two equations are given and we have to find the values of variable which may satisfy both the given equations, like cos = cos and sin = sin so the common solution is = n +, n I Similarly, sin = sin and so the common solution is, tan = tan = n +, n I Rule : Find the common values of between 0 and and then add n to this common value a c b
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