सम्मिश्र संख्या `15("cos"150^(@) + i"sin"150^(@))` का सम्मिश्र संख्या `3("cos"60^(@) + i"sin"60^(@))` से भागफल है

Options

A. `5i`
B. `5("cos" 120^(@) + "i sin"120^(@))`
C. `5("cos"120^(@) + "i sin"210^(@))`
D. 5

Answer

A

Explanation

`(15("cos"150^(@) + "i sin"150^(@)))/(3("cos"60^(@) + "i sin"60^(@)))`
` = 5 ("cos" 150^(@) + "i sin"150^(@)) ("cos"60^(@) + "i sin"60^(@))^(-1)`
` = 5("cos" 150^(@) + "i sin"150^(@)) ["cos" (-60^(@)) + "i sin" (-60^(@))] " " [because "cos" (-theta) = "cos" theta` तथा `"sin" (-theta) = -"sin" theta]`
`= 5("cos" 150^(@) "cos" 60^(@) -i^(2)"sin"60^(@)"sin"150^(@) + "i sin"150^(@)"cos"60^(@)) - "i cos"150^(@)"sin"60^(@)`
` = 5("cos" 150^(@)"cos"60^(@) + "sin"60^(@) "sin" 150^(@)) + i("sin" 150^(@)"cos"60^(@) ="cos" 150^(@)"sin"60^(@))`
` = 5["cos"(150^(@)-60^(@)) + "i sin"(150^(@)-60^(@)]`
` = 5("cos"90^(@) + "i sin"90^(@))`
` = 5(0 + i xx 1) = 5i`