用三角比公式化简:sin50'(1+根号3*tan10')

1个回答

  • 由题意得:

    方法1:sin50°(1+√3×tan 10°)

    =(sin50°cos10°+√3sin50°sin10°)/cos10°

    =[sin(60°-10°)cos10°+√3sin(60°-10°)sin10°]/cos10°

    =(sin60°cos10°²-cos60°sin10°cos10°+√3sin60°cos10°sin10°-√3cos60°sin10°²)/cos10°

    =[√3/2*(cos10°²-sin10°²)+sin10°cos10°]/cos10°

    =(√3/2×cos20°+1/2×sin20°)/cos10°

    =(sin60°cos20°+cos60°sin20°)/cos10°

    =sin80°/cos10°

    =1

    方法2:sin50°(1+√3tan10°)

    =sin50°[1+(√3sin10°/cos10°)]

    =sin50°[cos10°+√3sin10°)/cos10°]

    =2sin50°[(1/2)cos10°+(√3/2)sin10°)/cos10°]

    =2sin50°[cos60cos10°+sin60sin10°)/cos10°]

    =2sin50°cos(60°-10°)/cos10 °

    =2sin50°cos50°/cos10 °

    =sin100°/cos10 °

    =cos10°/cos10 °

    =1

    因此sin50°(1+√3×tan 10°) 的值为1