VIc
Lv 7
VIc asked in 科學及數學數學 · 1 decade ago

a-maths!!!

Find all the angles between 0° and 360° which satisfy the equation

tan2z=8/tanz

2 Answers

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  • 1 decade ago
    Favorite Answer

    As follows:

    By the formula: tan 2A = 2 tan A / ( 1 - tan^2 A ),

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    2007-12-17 22:50:11 補充:

    Something wrong with the answer below. There should NOT be the solutions 90* and 270*. They are only true for L.H.S., i.e. tan 180* and tan 540* are 0 but for R.H.S., they are not valid. tan 90* and tan 270* are undefined and so 8 / undefinded cannot be defined as well.

    2007-12-18 13:36:56 補充:

    tan ( A + B ) = ( tan A + tan B ) / ( 1 - tan A tan B ), sotan ( A + A ) = ( tan A + tan A ) / ( 1 - tan^2 A )and here comes the formula: tan 2A = 2 tan A / ( 1 - tan^2 A )

    2007-12-19 22:56:31 補充:

    'If 1/tanz is replaced by cotz, then 90 and 270 degrees are solutions.'It is still wrong becuase 1 / tan z = cot z, if 1 / tan 90* and 1 / tan 270* do not satisfy the equation, then how come cot 90* and cot 270* can satisfy the equation?

    Source(s): My Maths Knowledge
  • 1 decade ago

    Since sin2z=2sinzcosz and cos2z=square(cosz)-square(sinz) and

    tanz=sinz/cosz

    the equation can be converted to:

    sinzsin2z=8coszcos2z

    then

    2*square(sinz)*cosz=8*cube(cosz)-8cosz*square(sinz)

    then by rearranging the terms

    5*square(sinz)*cosz=4*cube(cosz)

    cosz(5*square(sinz)-4*square(cosz))=0

    cosz=0 or tanz=plus or minus(2/squareroot(5))

    cosz=0 implies that z= 90 degrees or 270 degrees

    tanz=plus or minus(2/squareroot(5)) implies that

    z=41.810 or 138.190 or 221.810 or 318.190 degrees, so there are six solutions

    2007-12-17 22:26:51 補充:

    the two solution 90 degrees and 270 degrees are based on the fact that at these angles,tanz tends to infinity, so 1/tanz is zero (1/tanz=cosz/sinz, cos90degree=cos270degree=0, sin90degree=1,sin270degree= -1)also tan2z are zero (tan180degree and tan 540degree are zero)

    2007-12-19 18:37:49 補充:

    Sorry, I have made a mistake.cotz= cosz/sinzcotz =0 at 90 and 270 degrees,while 1/tanz is undefined at these angles.so 90 and 270 are not solutions for this equation.However, if 1/tanz is replaced by cotz, then 90 and 270 degrees are solutions.

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