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Find the quotient 3(cos (5pi/12) + i sin (5pi/12) / 6(cos (pi/12) i sin (pi/12)). Express the quotient in rectangular form.

I have no idea what I did but I got (1/4) - 433i/1000.

  • Math -

    Using the identity
    cos(x)+i sin(x) = eix
    the expression simplifies considerably:
    3(cos (5pi/12) + i sin (5pi/12) / 6(cos (pi/12) + i sin (pi/12))
    =3(e5iπ/12)/6(eiπ/12)
    =(1/2)e(5iπ-iπ)/12
    =(1/2)eiπ/3
    =(1/2)(cos(π/3)+i sin(π/3))
    =(1/2)(1/2+(√3/3)i)
    =1/4+(√3/6)i

    Note: the identity can be derived by the expansion of eix
    =1 + ix + (ix)²/2! + (ix)³/3! + ...
    =1 +ix -x²/2! - x³/3! + ...
    =1 -x²/2! + x⁴/4! - x⁶/6! + ...
    + i( x - x³/3! + x⁵/5! - ...)
    = cos(x) + i sin(x)

    Alternatively, multiply both numerator and denominator by the conjugate of the denominator, namely (cos (pi/12)- i sin (pi/12))
    to reduce the denominator to:
    6(cos (pi/12)+i sin (pi/12))(cos (pi/12)-i sin (pi/12))
    =6(cos²(π/12)+sin²(π/12))
    =6
    The numerator becomes
    3(cos (5pi/12) + i sin (5pi/12)(cos (pi/12)-i sin (pi/12))
    =3(cos(5&pi/12)cos(&pi/12)+sin(5&pi/12)sin(&pi/12))
    + 3i(sin(5&pi/12)cos(&pi/12)-cos(5π/12)sin(&pi/12))
    =3(cos(&pi/3)+isin(&pi/3))
    So the result is also
    3(cos(&pi/3)+isin(&pi/3))/6
    =(1/2)(1/2+(√3/3)i)
    =1/4+(√3/6)i

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