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Консультация № 176506
05.02.2010, 05:46
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Уважаемые эксперты. Помогите пожалуйста решить уравнение
sin^2(2x)+sin^2(4x)=1-cos2x/cos3x
sin^2(2x)+sin^2(4x)=1-cos2x/cos3x
Обсуждение
05.02.2010, 20:04
общий
STASSY:
Здравствуйте!
Уточните, пожалуйста, вид правой части уравнения: (1 - cos 2x)/cos 3x или 1 - ((cos 2x)/cos 3x).
С уважением.
Здравствуйте!
Уточните, пожалуйста, вид правой части уравнения: (1 - cos 2x)/cos 3x или 1 - ((cos 2x)/cos 3x).
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
07.02.2010, 15:46
общий
это ответ
Здравствуйте, STASSY.
sin2 2x + sin2 4x = 1 – (cos 2x)/(cos 3x),
1 – cos2 2x + 4sin2 2x ∙ cos2 2x = 1 – (cos 2x)/(cos 3x),
-cos2 2x + 4sin2 2x ∙ cos2 2x = -(cos 2x)/(cos 3x),
cos2 2x – 4sin2 2x ∙ cos2 2x = (cos 2x)/(cos 3x),
cos2 2x ∙ cos 3x(1 – 4sin2 2x) = cos 2x,
cos 2x ∙ cos 3x(1 – 4sin2 2x) = 1,
cos 2x ∙ cos 3x(4cos2 2x – 3) = 1,
но, как известно, cos 3α = 4cos3 α – 3cos α,
поэтому 4cos2 α – 3 = (cos 3α)/(cos α), 4cos2 2x – 3 = (cos 6x)/(cos 2x),
(cos 2x ∙ cos 3x ∙ cos 6x)/(cos 2x) = 1,
cos 3x ∙ cos 6x = 1. (1)
Поскольку |cos 3x| ≤ 1, |cos 6x| ≤ 1, то равенство (1) может выполняться в двух случаях:
1) cos 3x = 1 и cos 6x = 1, откуда
x = 2πk/3, k – целое число, (2)
или
2) cos 3x = -1 и cos 6x = -1, откуда
x = π(2k + 1)/3, k – целое число. (3)
Следовательно, решением заданного уравнения является объединение решений (2) и (3), т. е.
x = 2πk/3, x = π(2k + 1)/3, k – целое число,
или
x = πm/3, m – целое число.
С уважением.
sin2 2x + sin2 4x = 1 – (cos 2x)/(cos 3x),
1 – cos2 2x + 4sin2 2x ∙ cos2 2x = 1 – (cos 2x)/(cos 3x),
-cos2 2x + 4sin2 2x ∙ cos2 2x = -(cos 2x)/(cos 3x),
cos2 2x – 4sin2 2x ∙ cos2 2x = (cos 2x)/(cos 3x),
cos2 2x ∙ cos 3x(1 – 4sin2 2x) = cos 2x,
cos 2x ∙ cos 3x(1 – 4sin2 2x) = 1,
cos 2x ∙ cos 3x(4cos2 2x – 3) = 1,
но, как известно, cos 3α = 4cos3 α – 3cos α,
поэтому 4cos2 α – 3 = (cos 3α)/(cos α), 4cos2 2x – 3 = (cos 6x)/(cos 2x),
(cos 2x ∙ cos 3x ∙ cos 6x)/(cos 2x) = 1,
cos 3x ∙ cos 6x = 1. (1)
Поскольку |cos 3x| ≤ 1, |cos 6x| ≤ 1, то равенство (1) может выполняться в двух случаях:
1) cos 3x = 1 и cos 6x = 1, откуда
x = 2πk/3, k – целое число, (2)
или
2) cos 3x = -1 и cos 6x = -1, откуда
x = π(2k + 1)/3, k – целое число. (3)
Следовательно, решением заданного уравнения является объединение решений (2) и (3), т. е.
x = 2πk/3, x = π(2k + 1)/3, k – целое число,
или
x = πm/3, m – целое число.
С уважением.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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