Question

How do you evaluate `e^( ( pi)/12 i) - e^( ( 13 pi)/8 i)` using trigonometric functions?

Answer 1

`=0.58+0.38i`

Explanation:

Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,
`e^{ix} = cos x + isin x`
using this formula we have

`e^{ipi/12} -e^{i13pi/12}`
`= cos (pi/12)+ isin (pi/12)-cos (13pi/8)- isin (13pi/8)`
`= cos (pi/12)+ isin (pi/12)-cos (pi+5pi/8)- isin (pi+5pi/8)`
`= cos (pi/12)+ isin (pi/12)+cos (5pi/8)+isin (5pi/8)`
`=0.96-0.54i-0.38+0.92i=0.58+0.38i`