Question

不通过求值,比较下列各组中两个三角函数值的大小sin(-53π/7)与sin(-59π/8),cos500°与cos530°

Answer 1

解：∵sin(-53π/7)=sin(-8π+3π/7)
=sin(π/2-π/14)
=cos(π/14)
sin(-59π/8)=sin(-8π+5π/8)
=sin(π/2+π/8)
=cos(π/8)
又∵cosx在x∈[0,π]上是减函数，
而π/14<π/8，
∴cosπ/14>cosπ/8
∴sin(-53π/7)>sin(-59π/8)
∵cos500°=cos(360°+90°+50°）
=cos(90°+50°）
=-sin50°
cos530°=cos(2×360°-170°）
=cos170°
=cos(90°+80°）
=-sin80°
又∵sin80°>sin50°
∴-sin80°<-sin50°
∴cos500°>cos530°

Answer 2

sin(-53π/7)=sin(3π/7) sin(-59π/8)= sin(5π/8)
所以sin(-53π/7)大于sin(-59π/8)
cos500°=-sin50°，cos530°=-sin80°
-sin80°<-sin50°
∴cos500°>cos530°

Answer 3

sin(-53π/7)=sin(3π/7) sin(-59π/8)= sin(5π/8)
sin(5π/8) = sin(3π/8) 小于sin(3π/7)
所以sin(-53π/7)大于sin(-59π/8)
,cos500°与cos530°（同理）