Question

已知x是锐角且sinx+cosx=7/5,则根号sinx+根号cosx的值

Answer 1

sin²x+cos²x = 1
(sinx+cosx)² = sin²x+cos²x +2sinxcosx = 47/25
所以2sinxcosx = 22/25
sinxcosx = 12/25 (解到这其实能猜出来是3/5 和 4/5)
(√sinx + √cosx)² = sinx+cosx +2√(sinxcosx)
= 7/5+2根号下(12/25)
= 7/5 + 2√12/5(√12先不要最简化)
= 1/5×[3+2√(3×4)+4]
= 1/5×[(√3)² + 2×√3×2+2²)]
= 1/5×(2+√3)²
所以√sinx + √cosx = (2+√3)/√5 = (2√5+√15)/5 ..

Answer 2

(根号sinx+根号cosx)^2 = sinx+cosx + 根号2 * 根号[( sinx+cosx)^2-1 ]

= 7/5 + ( 4* 根号3 ) / 5

所以：
根号sinx+根号cosx = ( 2+根号3) / 根号5