Question

已知点A，B的极坐标分别为（3，π/4)和（-3，π/12),则A和B之间的距离等于_________

Answer 1

直接套公式
★极坐标中两点距离公式
设： A(ρ1,θ1) B(ρ2,θ2)
|AB|=√[(ρ1)²+(ρ2)²-2·ρ1·ρ2·
cos(θ1-θ2)]
|AB|=√[3²+(-3)²-2×3×(-3)×cos(π/12-π/4)]
=√(18+18cosπ/6)
=√(18+9√3)
=√[9/2·(4+2√3)]
=√(9/2)·√[(√3+1)²]
=3√2/2·（√3+1）
=(3√6+3√2)/2

Answer 2

A坐标为：x=3cosπ/4,y=3sinπ/4

B坐标为:x=-3cosπ/12,y=-3sinπ/12

AB=√[(3cosπ/4+3cosπ/12)^2+(3sinπ/4+3sinπ/12)^2]
=3√(2+√3)

追问

答案为：（3倍根号6+3倍根号2）/2

追答

一样的
3√(2+√3) ＝3√[(4+2√3)/2]＝3√[(1+√3)^2/2]＝ 3(1+√3)/√2＝3(√6＋√2)/2

追问

“AB=√[(3cosπ/4+3cosπ/12)^2+(3sinπ/4+3sinπ/12)^2]
=3√(2+√3) ”这一步如何展开呢？请指教，谢谢！

追答

sinπ/12＝sin(π/3 -π/4)=sinπ/3cosπ/4-cosπ/3sinπ/4=(√6-√2)/4
同理
cosπ/12＝(√6+√2)/4
代入计算就可以了

Answer 3

you wen ti