IOAN ŞERDEAN, DANIEL SITARU

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1 Romanian Mathematical Magazine Web: The Author: This article is published with open access. TRIGONOMETRIC SUBSTITUTIONS IN PROBLEM SOLVING PART IOAN ŞERDEAN, DANIEL SITARU Abstract. In this paper are indicated a few useful trigonometric substitutions for solving problems. Solved problems are also a part of this article. a = y z, b = z x, c = x y s = x y z, S = xyzx y z) x y)y z)z x) xyz R = 4 xyzx y z), r = x y z x y z)yz x y z)zx x y z)xy r a = ; r b = ; r c = x y z sin A = yz x y)x z) ; sin B = zx y z)y x) ; sin C = xy z x)z y) cos A = x y z)x x y)x z) ; cos B = x y z)y y x)y z) ; = x y z)z z x)z y) tan A = yz x ; tan B = Key words and phrases. Trigonometric substitutions. zx y ; tan C = xy z

2 IOAN ŞERDEAN, DANIEL SITARU cot A x = yz ; cot B y = xz ; cot C z = zx tan A tan B tan C = xyz; cot A cot B cot C = xyz xx y z) yz yx y z) xz cos A = ; cos B = x y)x z) y z)y x) zx y z) xy = z x)z y) sin A = xyzx y z), sin B = xyzx y z) x y)x z) y z)y x) sin C = xyzx y z) z x)z y) Problem 4. If x, y, z > 0 then: xy z x)z y) yz x y)x z) zx y z)y x) 5 cos A = sin A = yz x y)x z) yz x y)x z) = cos A yz x y)x z) = cos A; xy z x)z y) = ; xz y x)y z) = cos B Inequality to prove can be written: ) ) cos A ) cos B 5 ) cos A cos B 5 7 cos A ) cos B 5 A C cos B = cos cos A = sin B cos A C By ) : 7 sin B cos A C B sin 5 sin B sin B cos A C sin B cos A C sin B cos A C ) cos A C sin B 0 A C 9 cos cos B = ) to prove) 0 ) 9 A C 9 cos 9 0

3 TRIGONOMETRIC SUBSTITUTIONS sin B cos A C ) A C 9 sin 9 0 Problem 44. If a, b, c > 0; a b c = then: c bc ca a bc b ca ) c c ) c )a b c) ca cb c a b c) ca cb c c a)c b) Donde: x = a b; y = b c; z = c a; s = a b c By ) : s y)s z) yz to prove) f : 0, π) R; fx) = sin x ; sin A = Problem 45. If x, y, z > 0 then: f x) = cos x ; f x) = 4 sin x 0 By Jensen: A B C ) fa) fb) fc) f sin A sin B sin C sin π = s y)s z) ; sin B s x)s z) yz = ; sin C xz = s x)s y) xy xy z) yz x) zx y) x y)y x)z x) x y z D. Grinberg)

4 4 IOAN ŞERDEAN, DANIEL SITARU xx y z) x y)x z) Inequality to prove becomes: yx y z) y z)y x) zx y z) z x)z y) a = x y; b = y z; c = z x; s = x y z ss b) ss c) ss a) ac bc cos A cos B A = π A, B = π B, C = π C cos π A cos π B cos π C sin A sin B sin C By Jordan s inequality: sin A A π ; sin B B π ; sin C C π By adding: sin A sin B sin C A B C ) = π π π = Problem 46. If a, b, c > 0; a b c = c then: a b c A. Nicolaescu; C. Pătraşcu) a = tan A; b = tan B; c = tan C; A, B, C 0, π ) tan A sin A sin B sin C tan B sin A sin B sin C CBS {}}{ tan C ) sin A sin B sin C 9 Problem 47. If x, y, z > 0; x y z = xyz then: x 4 y 4 z 4 6 = George Apostolopoulos)

5 TRIGONOMETRIC SUBSTITUTIONS 5 Denote: a = tan A, b = tan B, c = tan C tan A tan B tan C 6 tan A tan B tan C 6 6 CBS {}}{ ) cos A cos B 9 = 6 Problem 48. If x, y, z > 0; x y z = xyz then: xy yz zx x y z Denote x = tan A, y = tan B, z = tan C tan A tan B tan B tan C tan C tan A tan A tan B ) tan B tan C ) tan C tan A ) sin A sin B cos A cos B cos A cos B sin B sin C cos B cos B sin C sin A cos A cos A cosa B) cosb C) cosc A) cos A cos B cos B cos A cos A cos B cos A cos B cos B cos A cos A cos B cos C cos B cos A cos A cos B cos A cos B) cos B ) cos A) 0 Problem 49. If x, y, z > 0, xy yz zx = then: x x y y z z C. Popescu)

6 6 IOAN ŞERDEAN, DANIEL SITARU Denote x = tan A ; y = tan B ; z = tan C tan A tan A tan B tan B tan C tan C cos A cos B Problem 50. If a, b, c > 0; a b c = then: a b c c Denote a = xy; b = yz; c = zx; a b c = xy yz zx = For x = tan A ; y = tan B ; z = tan C A, B, C 0, π ) x y y z z x xyz xy yz zx) xyzx y z) xyz xyzx y z) xyz x y z tan A tan B tan C Problem 5. If x, y, z > ; x y z = then: x y z x y z Denote x = a, y = b, z = c, a, b, c > 0 x y = bc ca c = z a b c a b c For = sin A ; bc = B sin ; ca = C sin ; A, B, C 0, π ) the constraint can be written: sin A sin B sin C sin A sin B sin C = By squaring inequality to prove becomes: bc ca

7 TRIGONOMETRIC SUBSTITUTIONS 7 sin A sin B sin C Proposed problems 5. If a, b R; 5a b 7 = then: a b b a) Use: a = R sin p; b = R cos p) 5. If a, b R; a ; b then: a b Use: a = cos p ; b = [0, cos q ; p, q π ] [ π, π 54. If x ; y then: y x 4 y xy 6 Use: x = cos p ; y = 0, cos q ; p, q π ) ) 55. If x, y, y, v R x y = u v = then: a. xu yv b. xv yu c. x y)u v) x y)u v) d. x y)u v) x y)u v) ) Use: x = cos a; y = sin a; u = cos b; v = sin b; a, b 0, π) 56. If a, b R then: a b) 4 8a 4 b 4 ) a. a b) 4 8a 4 b 4 ) b. a b) 6 a 6 b 6 ) Use: tan x = b a ; x π, π ) ) 57. If x, y R; xy 0 then: x x 4y) x 4y Use: x = y tan p; p π, π ) ) 58. If x, y R; 6x 6y = 9 then: 5 5 y x Use: x = cos p; y = 4 ) sin p; p [0, π] ) )

8 8 IOAN ŞERDEAN, DANIEL SITARU 59. If x, y > 0; x 4y = 5 then x y ) Use: sin p = 5 ; cos p = If a, b R; 4a 9b = 5 then: 6a b 5 Use: 5 a = sin p; 5 b = cos p; p [0, π] ) 6. If x, y, a, b, c > 0, ax by = 0, a b = c then: x y Use: ac = cos p; bc = sin p; p [0, π) ) 6. If a > b > c > 0 then: ca c) cb c) c a c c Use: = sin p; = cos p; a b c b a b [ = cos v; u, v 0, π ] ) = sin v; TO BE CONTINUED! Mathematics Department, Theodor Costescu National Economic College, Drobeta Turnu - Severin, MEHEDINTI. address:

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