How do you evaluate $e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i}$ $e^( ( 11 pi)/6 i) - e^( ( 17 pi)/8 i)$ using trigonometric functions?

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How do you evaluate $e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i}$ $e^( ( 11 pi)/6 i) - e^( ( 17 pi)/8 i)$ using trigonometric functions?

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Answer 1 · 2018-03-03T13:52:21

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i} = 0.884 e^{- 1.505 i}$ $e^((11pi)/6i)-e^((17pi)/8i)=0.884e^(-1.505i)$

Explanation:

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i} = x + i y$ $e^((11pi)/6i)-e^((17pi)/8i)=x+iy$

$e^{\frac{11 π}{6} i} = cos (\frac{11 π}{6}) + i sin (\frac{11 π}{6})$ $e^((11pi)/6i)=cos((11pi)/6)+isin((11pi)/6)$

$e^{\frac{17 π}{8} i} = cos (\frac{17 π}{8}) + i sin (\frac{17 π}{8})$ $e^((17pi)/8i)=cos((17pi)/8)+isin((17pi)/8)$

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i} = (cos (\frac{11 π}{6}) + i sin (\frac{11 π}{6})) - (cos (\frac{17 π}{8}) + i sin (\frac{17 π}{8}))$ $e^((11pi)/6i)-e^((17pi)/8i)=(cos((11pi)/6)+isin((11pi)/6))-(cos((17pi)/8)+isin((17pi)/8))$

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i}$ $e^((11pi)/6i)-e^((17pi)/8i)$
$= (cos (\frac{11 π}{6}) - cos (\frac{17 π}{8}))$ $=(cos((11pi)/6)-cos((17pi)/8))$
$+ i (sin (\frac{11 π}{6}) - sin (\frac{17 π}{8}))$ $+i(sin((11pi)/6)-sin((17pi)/8))$

$cos (\frac{11 π}{6}) - cos (\frac{17 π}{8}) = - 2 sin (\frac{\frac{11 π}{6} + \frac{17 π}{8}}{2}) sin (\frac{\frac{11 π}{6} - \frac{17 π}{8}}{2})$ $cos((11pi)/6)-cos((17pi)/8)=-2sin(((11pi)/6+(17pi)/8)/2)sin(((11pi)/6-(17pi)/8)/2)$

$\frac{11 π}{6} + \frac{17 π}{8} = \frac{8 \times 11 π}{8 \times 6} + \frac{6 \times 17 π}{6 \times 8}$ $(11pi)/6+(17pi)/8=(8xx11pi)/(8xx6)+(6xx17pi)/(6xx8)$

$= \frac{88 π}{48} + \frac{102 π}{48} = \frac{(88 + 102) π}{48}$ $=(88pi)/48+(102pi)/48=((88+102)pi)/48$

$\frac{11 π}{6} + \frac{17 π}{8} = \frac{190 π}{48}$ $(11pi)/6+(17pi)/8=(190pi)/48$
$\frac{11 π}{6} + \frac{17 π}{8} = \frac{95 π}{24}$ $(11pi)/6+(17pi)/8=(95pi)/24$

$\frac{11 π}{6} - \frac{17 π}{8} = \frac{8 \times 11 π}{8 \times 6} - \frac{6 \times 17 π}{6 \times 8}$ $(11pi)/6-(17pi)/8=(8xx11pi)/(8xx6)-(6xx17pi)/(6xx8)$

$= \frac{88 π}{48} - \frac{102 π}{48} = \frac{(88 - 102) π}{48}$ $=(88pi)/48-(102pi)/48=((88-102)pi)/48$

$\frac{11 π}{6} - \frac{17 π}{8} = \frac{- 14 π}{48}$ $(11pi)/6-(17pi)/8=(-14pi)/48$
$\frac{11 π}{6} - \frac{17 π}{8} = \frac{- 7 π}{24}$ $(11pi)/6-(17pi)/8=(-7pi)/24$

$\frac{1}{2} (\frac{11 π}{6} + \frac{17 π}{8}) = \frac{95 π}{48}$ $1/2((11pi)/6+(17pi)/8)=(95pi)/48$

$\frac{1}{2} (\frac{11 π}{6} - \frac{17 π}{8}) = \frac{- 7 π}{48}$ $1/2((11pi)/6-(17pi)/8)=(-7pi)/48$

$- 2 sin (\frac{\frac{11 π}{6} + \frac{17 π}{8}}{2}) sin (\frac{\frac{11 π}{6} - \frac{17 π}{8}}{2}) = - 2 sin (\frac{95 π}{48}) sin (\frac{- 7 π}{48})$ $-2sin(((11pi)/6+(17pi)/8)/2)sin(((11pi)/6-(17pi)/8)/2)=-2sin((95pi)/48)sin((-7pi)/48)$

$cos (\frac{11 π}{6}) - cos (\frac{17 π}{8}) = - 2 sin (\frac{95 π}{48}) sin (\frac{- 7 π}{48})$ $cos((11pi)/6)-cos((17pi)/8)=-2sin((95pi)/48)sin((-7pi)/48)$

$sin (\frac{11 π}{6}) - sin (\frac{17 π}{8}) = 2 cos (\frac{\frac{11 π}{6} + \frac{17 π}{8}}{2}) sin (\frac{\frac{11 π}{6} - \frac{17 π}{8}}{2})$ $sin((11pi)/6)-sin((17pi)/8)=2cos(((11pi)/6+(17pi)/8)/2)sin(((11pi)/6-(17pi)/8)/2)$

$\frac{1}{2} (\frac{11 π}{6} + \frac{17 π}{8}) = \frac{95 π}{48}$ $1/2((11pi)/6+(17pi)/8)=(95pi)/48$

$\frac{1}{2} (\frac{11 π}{6} - \frac{17 π}{8}) = \frac{- 7 π}{48}$ $1/2((11pi)/6-(17pi)/8)=(-7pi)/48$

$sin (\frac{11 π}{6}) - sin (\frac{17 π}{8}) = 2 cos (\frac{95 π}{48}) sin (\frac{- 7 π}{48})$ $sin((11pi)/6)-sin((17pi)/8)=2cos((95pi)/48)sin((-7pi)/48)$

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i}$ $e^((11pi)/6i)-e^((17pi)/8i)$
$= (cos (\frac{11 π}{6}) - cos (\frac{17 π}{8}))$ $=(cos((11pi)/6)-cos((17pi)/8))$
$+ i (sin (\frac{11 π}{6}) - sin (\frac{17 π}{8}))$ $+i(sin((11pi)/6)-sin((17pi)/8))$

$cos (\frac{11 π}{6}) - cos (\frac{17 π}{8}) = - 2 sin (\frac{95 π}{48}) sin (\frac{- 7 π}{48})$ $cos((11pi)/6)-cos((17pi)/8)=-2sin((95pi)/48)sin((-7pi)/48)$
$sin (\frac{11 π}{6}) - sin (\frac{17 π}{8}) = 2 cos (\frac{95 π}{48}) sin (\frac{- 7 π}{48})$ $sin((11pi)/6)-sin((17pi)/8)=2cos((95pi)/48)sin((-7pi)/48)$

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i} = - 2 sin (\frac{95 π}{48}) sin (\frac{- 7 π}{48}) + i (2 cos (\frac{95 π}{48}) sin (\frac{- 7 π}{48}))$ $e^((11pi)/6i)-e^((17pi)/8i)=-2sin((95pi)/48)sin((-7pi)/48)+i(2cos((95pi)/48)sin((-7pi)/48))$

$\frac{95 π}{48} = 6.218$ $(95pi)/48=6.218$
$\frac{7 π}{48} = 0.458$ $(7pi)/48=0.458$

$cos (\frac{95 π}{48}) = 0.998$ $cos((95pi)/48)=0.998$
$sin (\frac{95 π}{48}) = - 0.0654$ $sin((95pi)/48)=-0.0654$
$cos (\frac{- 7 π}{48}) = 0.897$ $cos((-7pi)/48)=0.897$
$sin (\frac{- 7 π}{48}) = - 0.442$ $sin((-7pi)/48)=-0.442$

$- 2 (- 0.0654) \times (- 0.442) + i (2 (0.998) \times (- 0.442)$ $-2(-0.0654)xx(-0.442)+i(2(0.998)xx(-0.442)$

$2 \times 0.0654 \times 0.442 - 2 \times 0.998 \times 0.442 i$ $2xx0.0654xx0.442-2xx0.998xx0.442i$

$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i} = 0.058 - 0.882 i$ $e^((11pi)/6i)-e^((17pi)/8i)=0.058-0.882i$

$r = \sqrt{{0.058}^{2} + {(- 0.882)}^{2} = 0.884}$ $r=sqrt(0.058^2+(-0.882)^2=0.884$

$θ = \frac{{tan}^{- 1} (- 0.882)}{0.058} = - 1.505$ $theta=tan^-1(-0.882)/(0.058)=-1.505$

$0.058 - 0.882 i = 0.884 (cos (- 1.505) + i sin (- 1.505))$ $0.058-0.882i=0.884(cos(-1.505)+isin(-1.505))$

$cos (- 1.505) + i sin (- 1.505) = e^{(- 1.505) i}$ $cos(-1.505)+isin(-1.505)=e^((-1.505)i$
$e^{\frac{11 π}{6} i} - e^{\frac{17 π}{8} i} = 0.884 e^{- 1.505 i}$ $e^((11pi)/6i)-e^((17pi)/8i)=0.884e^(-1.505i)$

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