cos2a+cos4a+cos6a-cos8a = ?

[did5]

cos2α+cos4α+cos6α-cos8α=

[mail_dinko]

[tex]cos2\alpha +cos4\alpha +cos6\alpha -cos8\alpha =[/tex]


[tex]cos2\alpha=cos^2 \alpha - sin^2 \alpha[/tex]


[tex]cos4\alpha=cos^2 2 \alpha - sin^2 2 \alpha=(cos^2 \alpha - sin^2 \alpha)^2-(2sin \alpha cos \alpha )^2=[/tex]
[tex]=cos^4 \alpha -2sin^2 \alpha cos^2 \alpha + sin^4 \alpha-4sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=(cos^2 \alpha)^2 +2sin^2 \alpha cos^2 \alpha + ( sin^2 \alpha)^2-6sin^2 \alpha cos^2 \alpha-2sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=(cos^2 \alpha + sin^2 \alpha)^2-8sin^2 \alpha cos^2 \alpha=1-8sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=1-8sin^2 \alpha (1-sin^2 \alpha)=1-8sin^2 \alpha + 8sin^4 \alpha[/tex]


[tex]cos6\alpha = cos^2 3 \alpha - sin^2 3 \alpha=(4cos^3 \alpha -3cos \alpha)^2-(3sin \alpha -4sin^3 \alpha)^2=[/tex]
[tex]=16cos^6 \alpha -24 cos^4 \alpha + 9 cos^2 \alpha-9sin^2 \alpha +24 sin^4 \alpha - 16 sin^6 \alpha=[/tex]
[tex]=16(cos^6 \alpha-sin^6 \alpha)-24 ( cos^4 \alpha-sin^4 \alpha)+9 (cos^2 \alpha - sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)(cos^4 \alpha +sin^2 \alpha cos^2 \alpha+ sin^4 \alpha)-[/tex]
[tex]-24 ( cos^2 \alpha-sin^2 \alpha)(cos^2 \alpha +sin^2 \alpha)+9 (cos^2 \alpha - sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)[(cos^2 \alpha)^2 +sin^2 \alpha cos^2 \alpha+ (sin^2 \alpha)^2+2sin^2 \alpha cos^2 \alpha-2sin^2 \alpha cos^2 \alpha]-[/tex]
[tex]-24 ( cos^2 \alpha-sin^2 \alpha)+9 (cos^2 \alpha - sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)[(cos^2 \alpha + sin^2 \alpha)^2 -sin^2 \alpha cos^2 \alpha]-15 ( cos^2 \alpha-sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)[1 -sin^2 \alpha cos^2 \alpha]-15 ( cos^2 \alpha-sin^2 \alpha)=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[16(1- sin^2 \alpha cos^2 \alpha)-15]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[16-16 sin^2 \alpha cos^2 \alpha-15]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[1-16 sin^2 \alpha cos^2 \alpha]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[1-16 sin^2 \alpha (1- sin^2 \alpha)]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[1-16 sin^2 \alpha + 16sin^4 \alpha]=[/tex]

[tex]cos 8 \alpha = cos^2 4 \alpha - sin^2 4 \alpha=[/tex]
[tex]=(1-8sin^2 \alpha + 8sin^4 \alpha)^2-4sin^2 2 \alpha cos ^2 2 \alpha=[/tex]
[tex]=(1-8sin^2 \alpha + 8sin^4 \alpha)^2-4sin^2 2 \alpha (1- sin ^2 2 \alpha)=[/tex]
[tex]=(1-8sin^2 \alpha + 8sin^4 \alpha)^2-4sin^2 2 \alpha + sin ^4 2 \alpha=[/tex]

Нямам време да го проверя и да го разпиша нацяло, пробвай сам(а)

[ganka simeonova]

[tex]cos2\alpha +cos4\alpha +cos6\alpha -cos8\alpha =[/tex]


[tex]cos2\alpha=cos^2 \alpha - sin^2 \alpha[/tex]


[tex]cos4\alpha=cos^2 2 \alpha - sin^2 2 \alpha=(cos^2 \alpha - sin^2 \alpha)^2-(2sin \alpha cos \alpha )^2=[/tex]
[tex]=cos^4 \alpha -2sin^2 \alpha cos^2 \alpha + sin^4 \alpha-4sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=(cos^2 \alpha)^2 +2sin^2 \alpha cos^2 \alpha + ( sin^2 \alpha)^2-6sin^2 \alpha cos^2 \alpha-2sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=(cos^2 \alpha + sin^2 \alpha)^2-8sin^2 \alpha cos^2 \alpha=1-8sin^2 \alpha cos^2 \alpha=[/tex]
[tex]=1-8sin^2 \alpha (1-sin^2 \alpha)=1-8sin^2 \alpha + 8sin^4 \alpha[/tex]


[tex]cos6\alpha = cos^2 3 \alpha - sin^2 3 \alpha=(4cos^3 \alpha -3cos \alpha)^2-(3sin \alpha -4sin^3 \alpha)^2=[/tex]
[tex]=16cos^6 \alpha -24 cos^4 \alpha + 9 cos^2 \alpha-9sin^2 \alpha +24 sin^4 \alpha - 16 sin^6 \alpha=[/tex]
[tex]=16(cos^6 \alpha-sin^6 \alpha)-24 ( cos^4 \alpha-sin^4 \alpha)+9 (cos^2 \alpha - sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)(cos^4 \alpha +sin^2 \alpha cos^2 \alpha+ sin^4 \alpha)-[/tex]
[tex]-24 ( cos^2 \alpha-sin^2 \alpha)(cos^2 \alpha +sin^2 \alpha)+9 (cos^2 \alpha - sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)[(cos^2 \alpha)^2 +sin^2 \alpha cos^2 \alpha+ (sin^2 \alpha)^2+2sin^2 \alpha cos^2 \alpha-2sin^2 \alpha cos^2 \alpha]-[/tex]
[tex]-24 ( cos^2 \alpha-sin^2 \alpha)+9 (cos^2 \alpha - sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)[(cos^2 \alpha + sin^2 \alpha)^2 -sin^2 \alpha cos^2 \alpha]-15 ( cos^2 \alpha-sin^2 \alpha)=[/tex]
[tex]=16(cos^2 \alpha-sin^2 \alpha)[1 -sin^2 \alpha cos^2 \alpha]-15 ( cos^2 \alpha-sin^2 \alpha)=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[16(1- sin^2 \alpha cos^2 \alpha)-15]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[16-16 sin^2 \alpha cos^2 \alpha-15]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[1-16 sin^2 \alpha cos^2 \alpha]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[1-16 sin^2 \alpha (1- sin^2 \alpha)]=[/tex]
[tex]= ( cos^2 \alpha-sin^2 \alpha)[1-16 sin^2 \alpha + 16sin^4 \alpha]=[/tex]

[tex]cos 8 \alpha = cos^2 4 \alpha - sin^2 4 \alpha=[/tex]
[tex]=(1-8sin^2 \alpha + 8sin^4 \alpha)^2-4sin^2 2 \alpha cos ^2 2 \alpha=[/tex]
[tex]=(1-8sin^2 \alpha + 8sin^4 \alpha)^2-4sin^2 2 \alpha (1- sin ^2 2 \alpha)=[/tex]
[tex]=(1-8sin^2 \alpha + 8sin^4 \alpha)^2-4sin^2 2 \alpha + sin ^4 2 \alpha=[/tex]

Нямам време да го проверя и да го разпиша нацяло, пробвай сам(а)

Kакво е това чудо?
Я групирай по два косинуса и приложи сбор и разлика на косинуси..

[mail_dinko]

Може ли жокер малък поне? Не схващам как да групирам, в смисъл не е като подобни едночлени, защото всичките 4 неща са различни, или греша

[mail_dinko]

Вече това, за което говорите не ми е "тъмна Индия", взехме тази седмица формулите за сбор на два синуса и т.н. Обаче след като ги групирам нищо не става. Пробвах първо с трето и първо с второ, ама нищо не ми излиза, нищо във вид на произведение

[ganka simeonova]

За да се групира във вид на произведение, последният знак трябва да е +.
Не всеки тригонометричен израз може да се представи като произведение:)

[mail_dinko]

Вземам го със знак плюс, за удобство ще сменя ъгъла с хикс вместо алфа
[tex]cos2x+cos4x+cos6x+cos8x=[/tex]
[tex]=2cos{ \frac{4x+6x}{2} }cos{ \frac {4x-6x}{2}}+2cos{ \frac {2x+8x}{2}}cos{ \frac {2x-8x}{2}}=[/tex]
[tex]=2cos{ \frac{\cancel{10}x}{\cancel {2}} }cos{ \frac {-\cancel{2}x}{\cancel {2}}}+2cos{ \frac {\cancel {10}x}{\cancel {2}}}cos{ \frac {-\cancel{6}x}{\cancel {2}}}=[/tex]
[tex]=2cos{5x }cos{ (-x)}+2cos{ 5x}cos{ (-3x)}=[/tex]
[tex]=2cos{5x }cos{ x}+2cos{ 5x}cos{ 3x}=[/tex]
[tex]=2cos{5x }(cos{ x}+cos{ 3x})=[/tex]
[tex]=2cos{5x }2cos{ \frac{x+3x}{2} }cos{ \frac {x-3x}{2}}=[/tex]
[tex]=2cos{5x }2cos{ \frac{\cancel{4}x}{\cancel {2}} }cos{ \frac {-\cancel{2}x}{\cancel {2}}}=[/tex]
[tex]=2cos{5x }2cos{2x }cos{ (-x)}=[/tex]
[tex]=4cos{5x }cos{2x }cos{ x}[/tex]

[ganka simeonova]