Question

一高一数学题：同角三角函数的基本关系。

若sinx+cosx=根号下2,那么(sin^4)x+(cos^4)x的值为( )

Answer 1

方法1: 因为((sin^2)x+(cos^2)x)^2=(sin^4)x+(cos^4)x+2*(sin^2)x (cos^2)x
由 (sinx+cosx)^2=(sin^2)x+(cos^2)x+2*sinx cosx，即
sinx cosx = ((sinx+cosx)^2-=(sin^2)x+(cos^2)x)/2 =(2-1)/2=1/2
所以(sin^4)x+(cos^4)x = ((sin^2)x+(cos^2)x)^2- 2*(sin^2)x (cos^2)x=1-2*(1/2)^2=1/2
方法2:取特值取sinx=cosx=√2/2,即(sin^4)x+(cos^4)x=(√2/2)^4+(√2/2)^4=1/2

Answer 2

sinx+cosx=√2,

(sinx+cosx)²=2
sin²x+cos²x+2sinxcosx=2
1+2sinxcosx=2
sinxcosx=1/2

(sin^4)x+(cos^4)x=(sin²x+cos²x)²-2sin²xcos²

=1-2*(1/4)
=1/2

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Answer 3

sinx+cosx=√2 得 (sinx+cosx)²=1+2sinxcosx=2 得 sinxcosx=1/2

(sin^4)x+(cos^4)x = (sin²x+cos²x)² -2sin²xcos²x = 1²- 2(1/2)² =1/2

Answer 4

因为sinx+cosx=根号下2所以sinx=cosx=二分之一根号下2 ,(sin^4)x+(cos^4)x=1/2