已知0小于a小于180,(1+cos2a)/[cot(a/2)-tan(a/2)],求sina+cosa

3个回答

  • (1+cos2a)/[cot(a/2)-tan(a/2)]

    =(1+cos2a)/[cos(a/2)/sin(a/2) - sin(a/2)/cos(a/2)]

    = (1+cos2a) sin(a/2)cos(a/2) / [cos(a/2)cos(a/2)-sin(a/2)sin(a/2)]

    =(2cosacosa-1+1) 1/2sina /cosa

    =sinacosa

    有sin^2a + cos^2a =1(sin和cos 的平方和 是1 )

    所以(sina+cosa )^2 =1+2sinacosa=1+3/5=8/5

    sina+cosa =根号下8/5(负值舍去)

    解题思路:切割化弦

    诱导公式

    sin和cos 的平方和 是1

    sin(-a)=-sin(a)

    cos(-a)=cos(a)

    sin(pi/2-a)=cos(a)

    cos(pi/2-a)=sin(a)

    sin(pi/2+a)=cos(a)

    cos(pi/2+a)=-sin(a)

    sin(pi-a)=sin(a)

    cos(pi-a)=-cos(a)

    sin(pi+a)=-sin(a)

    cos(pi+a)=-cos(a)

    tgA=tanA=sinA/cosA

    两角和与差的三角函数

    sin(a+b)=sin(a)cos(b)+cos(α)sin(b)

    cos(a+b)=cos(a)cos(b)-sin(a)sin(b)

    sin(a-b)=sin(a)cos(b)-cos(a)sin(b)

    cos(a-b)=cos(a)cos(b)+sin(a)sin(b)

    tan(a+b)=(tan(a)+tan(b))/(1-tan(a)tan(b))

    tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b))

    三角函数和差化积公式

    sin(a)+sin(b)=2sin((a+b)/2)cos((a-b)/2)

    sin(a)−sin(b)=2cos((a+b)/2)sin((a-b)/2)

    cos(a)+cos(b)=2cos((a+b)/2)cos((a-b)/2)

    cos(a)-cos(b)=-2sin((a+b)/2)sin((a-b)/2)

    积化和差公式

    sin(a)sin(b)=-1/2*[cos(a+b)-cos(a-b)]

    cos(a)cos(b)=1/2*[cos(a+b)+cos(a-b)]

    sin(a)cos(b)=1/2*[sin(a+b)+sin(a-b)]

    二倍角公式

    sin(2a)=2sin(a)cos(a)

    cos(2a)=cos^2(a)-sin^2(a)=2cos^2(a)-1=1-2sin^2(a)

    半角公式

    sin^2(a/2)=(1-cos(a))/2

    cos^2(a/2)=(1+cos(a))/2

    tan(a/2)=(1-cos(a))/sin(a)=sin(a)/(1+cos(a))

    万能公式

    sin(a)= (2tan(a/2))/(1+tan^2(a/2))

    cos(a)= (1-tan^2(a/2))/(1+tan^2(a/2))

    tan(a)= (2tan(a/2))/(1-tan^2(a/2))

    其它公式

    a*sin(a)+b*cos(a)=sqrt(a^2+b^2)sin(a+c) [其中,tan(c)=b/a]

    a*sin(a)-b*cos(a)=sqrt(a^2+b^2)cos(a-c) [其中,tan(c)=a/b]

    1+sin(a)=(sin(a/2)+cos(a/2))^2

    1-sin(a)=(sin(a/2)-cos(a/2))^2

    其他非重点三角函数

    csc(a)=1/sin(a)

    sec(a)=1/cos(a)

    双曲函数

    sinh(a)=(e^a-e^(-a))/2

    cosh(a)=(e^a+e^(-a))/2

    tgh(a)=sinh(a)/cosh(a)