化简y=(sinx)的平方+[sin(x+π/3)]的平方+[sin(x-π/3)]的平方,并求y的最大值

1个回答

  • y=﹙sinx)的平方+[sin(x+π/3)]的平方+[sin(x-π/3)]的平方

    sin(x﹢π/3)=sinxcosπ/3﹢cosxsinπ/3=sinx/2﹢√3cosx/2

    sin(x﹣π/3)=sinxcosπ/3﹣cosxsinπ/3=sinx/2﹣√3cosx/2

    [sin(x+π/3)]的平方=﹙sinx/2﹢√3cosx/2﹚²=﹙sin²x﹢3cos²x+2√3sinxsosx﹚/4

    [sin(x-π/3)]的平方=﹙sinx/2﹣√3cosx/2﹚²=﹙sin²x﹢3cos²x﹣2√3sinxsosx﹚/4

    ∴y=﹙sinx)的平方+[sin(x+π/3)]的平方+[sin(x-π/3)]的平方

    =sin²x+﹙sin²x﹢3cos²x﹚/2

    =3﹙sin²x﹢cos²x﹚/2

    =3/2

    ∴y的最大值=3/2