若cos(65°+α)=1/4,其中α为第三象限角,求cos(115°-α）+sin(α-115°）的值

Question

若cos(65°+α)=1/4,其中α为第三象限角,求cos(115°-α）+sin(α-115°）的值

Answer 1

cos(115º-α)=cos[180º-(65º+α)]=-cos(65º+α)=-1/4.
sin(α-115º)=-sin(115º-α)=-sin[180º-(65º+α)]=-sin(65º+α)
∵cos(65º+α)=1/4
∴cos²﹙65º+α﹚=1/16
又∵sin²(65º+α)+cos²(65º+α)=1
∴sin²(65º+α）=15/16
又∵α为第三象限角.（sin﹙65º+α﹚﹤0﹚
∴sin(65º+α)=-√15/4
∴cos(115º-α)+sin(α-115º)=(-1/4)+(-√15/4)=-(√15+1)/4