Question

How do you evaluate `e^( ( 13 pi)/8 i) - e^( ( 5 pi)/4 i)` using trigonometric functions?

Answer 1

Euler's theorem: `e^(itheta)=costheta-isintheta`

Explanation:

`e^((13pi)/8i)-e^((5pi)/4i)`
`=(cos((13pi)/8)+isin((13pi)/8))-(cos((5pi)/4)+isin((5pi)/4))`
`=sin(pi/8)-icos(pi/8)-1/sqrt(2)+1/sqrt(2)i`
`~=-0.32442 - 0.21677 i`

Answer 2

The answer is `=((sqrt(2-sqrt2))/2+sqrt2/2)-i((sqrt(2+sqrt2))/2+sqrt2/2)`

Explanation:

Reminder :

Euler's relation

`e^(itheta)=costheta+isintheta`

Here, we have

`z=e^(13/8pi)-e^(5/4pi)=cos(13/8pi)+isin(13/8pi)-cos(5/4pi)-isin(5/4pi)`

Therefore,

`13/8pi=5/8pi+pi=1/8pi+3/2pi`

`5/4pi=1/4pi+pi`

`cos(1/4pi)=1-2sin^2(1/8pi)=2cos^2(1/8pi)-1`

`sin(1/8pi)=sqrt((1-cos(1/4pi))/2)=sqrt((1-sqrt2/2)/(2))=(sqrt(2-sqrt2))/2`

`cos(1/8pi)=sqrt((1+cos(1/4pi))/2)=sqrt((1+sqrt2/2)/(2))=(sqrt(2+sqrt2))/2`#

So,
`z=cos(1/8pi+3/2pi)+isin(1/8pi+3/2pi)-cos(1/4pi+pi)-isin(1/4pi+pi)`

`cos(1/8pi+3/2pi)=cos(1/8pi)cos(3/2pi)-sin(1/8pi)sin(3/2pi)`

`=(sqrt(2+sqrt2))/2*0-(sqrt(2-sqrt2))/2*(-1)=(sqrt(2-sqrt2))/2`

`sin(1/8pi+3/2pi)=sin(1/8pi)cos(3/2pi)+cos(1/8pi)sin(3/2pi)`

`=(sqrt(2-sqrt2))/2*0+(sqrt(2+sqrt2))/2*(-1)=-(sqrt(2+sqrt2))/2`

`cos(1/4pi+pi)=cos(1/4pi)cos(pi)-sin(1/4pi)sin(pi)`

`=sqrt2/2*-1-sqrt2/2*0=-sqrt2/2`

`sin(1/4pi+pi)=sin(1/4pi)cos(pi)+cos(1/4pi)sin(pi)`

`=sqrt2/2*-1+sqrt2/2*0=sqrt2/2`

Finally,

`z=(sqrt(2-sqrt2))/2-i(sqrt(2+sqrt2))/2+sqrt2/2-isqrt2/2`

`=((sqrt(2-sqrt2))/2+sqrt2/2)-i((sqrt(2+sqrt2))/2+sqrt2/2)`