Question

How do you multiply `e^(( 11 pi )/ 8 i) * e^( pi/2 i )` in trigonometric form?

Answer 1

Euler formula `e^(itheta) = cos(theta)+isin(theta)` that would convert to trigonometric form. When we multiply trigonometric form we add the angles and multiply the modulus.

Explanation:

`e^((11pi)/8i)*e^((pi/2)i)`

`e^((11pi)/8i) = cos((11pi)/8) + isin((11pi)/8)`

`e^((pi/2)i) = cos(pi/2) + isin(pi/2)`

`e^((11pi)/8i)*e^((pi/2)i)`

`= (cos((11pi)/8) + isin((11pi)/8))*(cos(pi/2) + isin(pi/2))`

`=cos((11pi)/8+pi/2)+isin((11pi)/8+pi/2)`

`=cos((11pi)/8+(4pi)/8)+isin((11pi)/8+(4pi)/4)`

`=cos((11+4)pi/8) + isin((11+4)pi/8)`

`=cos((15pi)/8)+isin((15pi)/8)`

Note: The question said in trigonometric form so converted to trigonometric form and multiplied. Using Euler's form would be easy multiply and then convert, the choice is yours.

Alternate method:
`e^((11pi)/8i)*e^((pi/2)i)`
`=e^(((11pi)/8+pi/2)i)` using exponent rule `a^m*a^n=a^(m+n)`
`=e^(((15pi)/8)i)`

Use the Euler's formula

`=cos((15pi)/8) + isin((15pi)/8)` Answer.