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Консультация № 144341
18.09.2008, 21:28
0.00 руб.
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2
1
Здравствуйте, помогите пожалуйста доказать тригонометрическое тождество.
2arccos(квадратный_корень_из_(1+x)/2) = arccosx
2arccos(квадратный_корень_из_(1+x)/2) = arccosx
Обсуждение
Неизвестный
19.09.2008, 05:31
общий
это ответ
Здравствуйте, alex145! Вобще есть табличная формула удвоения для обратных тригонометических функий . В частности для арккосинуса имеем :
2*arccosx = arccos(2*(x^2)-1) при x>=0 ( больше или равно 0 ) ,
2*arccosx = Pi - arccos(2*(x^2)-1) при x<=0 .
Только в нашем случае вместо простого х имеем sqrt((х+1)/2) ...
Как известно arccos(-x)=Pi - arccosx и sqrt - корень квадратный .
Докажем случай при х>=0 .
2*arccos(sqrt((1+x)/2)) = arccos(2*((1+x)/2)-1) = arccos(1+x-1) = arccosx .
Так же докажем тождество и при х<=0 .
2*arccos(sqrt((1-x)/2)) = Pi - arccos(2*((1-x)/2)-1) = (2*Pi) - arccos(1-x-1) = (2*Pi) - arccos(-x) =
= Pi - Pi + arccosx = arccosx .
Доказательства крайне элементарные поэтому у меня возникают сомнения : может быть , Вас попросят
доказать , между делом , и формулу удвоения арккосинуса ?
Как Вы должны знать arccos(cosx) = x npи 0<=x<=Pi .
В нашем случае делаем замену х=соsу , тогда получим
2*arccos(cosy) = arccos(2*((cosy)^2)-1) = arccos(cos(2*y)) => 2*y = 2*y .
Это значит что тождество верное . При Pi<=x<=2*Pi доказательство такое же ввиду чётности косинуса .
2*arccosx = arccos(2*(x^2)-1) при x>=0 ( больше или равно 0 ) ,
2*arccosx = Pi - arccos(2*(x^2)-1) при x<=0 .
Только в нашем случае вместо простого х имеем sqrt((х+1)/2) ...
Как известно arccos(-x)=Pi - arccosx и sqrt - корень квадратный .
Докажем случай при х>=0 .
2*arccos(sqrt((1+x)/2)) = arccos(2*((1+x)/2)-1) = arccos(1+x-1) = arccosx .
Так же докажем тождество и при х<=0 .
2*arccos(sqrt((1-x)/2)) = Pi - arccos(2*((1-x)/2)-1) = (2*Pi) - arccos(1-x-1) = (2*Pi) - arccos(-x) =
= Pi - Pi + arccosx = arccosx .
Доказательства крайне элементарные поэтому у меня возникают сомнения : может быть , Вас попросят
доказать , между делом , и формулу удвоения арккосинуса ?
Как Вы должны знать arccos(cosx) = x npи 0<=x<=Pi .
В нашем случае делаем замену х=соsу , тогда получим
2*arccos(cosy) = arccos(2*((cosy)^2)-1) = arccos(cos(2*y)) => 2*y = 2*y .
Это значит что тождество верное . При Pi<=x<=2*Pi доказательство такое же ввиду чётности косинуса .
Неизвестный
19.09.2008, 05:35
общий
" 2*arccos(sqrt((1-x)/2)) = Pi - arccos(2*((1-x)/2)-1) = (2*Pi) - arccos(1-x-1) = (2*Pi) - arccos(-x) =
= Pi - Pi + arccosx = arccosx . "
Простите - правильно будет одно Пи , а не 2 ...
2*arccos(sqrt((1-x)/2)) = Pi - arccos(2*((1-x)/2)-1) = Pi - arccos(1-x-1) = Pi - arccos(-x) =
= Pi - Pi + arccosx = arccosx .
= Pi - Pi + arccosx = arccosx . "
Простите - правильно будет одно Пи , а не 2 ...
2*arccos(sqrt((1-x)/2)) = Pi - arccos(2*((1-x)/2)-1) = Pi - arccos(1-x-1) = Pi - arccos(-x) =
= Pi - Pi + arccosx = arccosx .
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