Question

已知sin(六分之π+a)=五分之三,三分之π<a<六分之五π,求cosa的值

Answer 1

解答：
三分之π<a<六分之五π
∴ π/2<a+π/6<π
∴ cos(a+π/6)<0
∵ cos²(a+π/6)=1-sin²(a+π/6)=1-(3/5)²=16/25
∴ cos(a+π/6)=-4/5
∴ cosa
= cos[(a+π/6)-π/6)]
= cos(a+π/6)cos(π/6)+sin(a+π/6)sin(π/6)
= (-4/5)*(√3/2)+(3/5)*(1/2)
= (-4√3+3)/10

Answer 2

∵π/3<a<(5/6)π

∴sina>0，cosa<0

sin(π/6+a)

=sin(π/6)cosa+cos(π/6)sina

=sin(π/6)cosa+cos(π/6)√(1-cos²a)

=3/5

∴cos(π/6)√(1-cos²a)=3/5-sin(π/6)cosa

∴cos²(π/6)(1-cos²a)=3^2/5^2-2*(3/5)*sin(π/6)cosa+sin²(π/6)cos²a

∴3/4(1-cos²a)=9/25-(3/5)cosa+(1/4)cos²a

∴3/4-(3/4)cos²a=9/25-(3/5)cosa+(1/4)cos²a

∴cos²a-(3/5)cosa-39/100=0

解方程得，cosa=(3+4√3)/10（不合题意，故舍去），或者，cosa=(3-4√3)/10

所以，cosa的值为(3-4√3)/10

Answer 3

π/3<a<5π/6
π/2<π/6+a<π
sin(π/6+a)=3/5
sinπ/6cosa+sinacosπ/6=3/5
1/2cosa+√3/2sina=3/5 ①
cos(π/6+a)=-4/5
cosπ/6cosa+sinπ/6sina=-4/5
√3/2cosa+1/2sina=-4/5 ②
①*1/2-②*√3/2
cosa=(2√3-3)/5