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Rasumasan
@Rasumasan
July 2022
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найдите значение выражения sin^3x+cos^3x, если sinx+cosx=a
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kirichekov
Verified answer
Sinx+cosx=a
sin³x+cos³x=(3a-a
³)/2
1. sin³x+cos³x=(sinx+cosx)*(sin²x-sinx*cosx+cos²x)=(sinx+cosx)*(1-sinx*cosx)
2. (sinx+cosx)²=a²
(sinx+cosx)²=sin²x+2sinx*cosx+cos²x=1+2sinx*cosx
1+2sinx*cosx=a²
sinx*cosx=(a²-1)/2
3. sin³x+cos³x=a*(1-(a²-1)/2)=a*(2-a²+1)/2=a*(3-a²)/2=(3a-a³)/2
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Answers & Comments
Verified answer
Sinx+cosx=asin³x+cos³x=(3a-a³)/2
1. sin³x+cos³x=(sinx+cosx)*(sin²x-sinx*cosx+cos²x)=(sinx+cosx)*(1-sinx*cosx)
2. (sinx+cosx)²=a²
(sinx+cosx)²=sin²x+2sinx*cosx+cos²x=1+2sinx*cosx
1+2sinx*cosx=a²
sinx*cosx=(a²-1)/2
3. sin³x+cos³x=a*(1-(a²-1)/2)=a*(2-a²+1)/2=a*(3-a²)/2=(3a-a³)/2