Question

How do you write the complex number in standard form `8(cos(pi/12)+isin(pi/12))`?

Answer 1

`8(cos(pi/12)+isin(pi/12))=(2sqrt6+2sqrt2)+i(2sqrt6-2sqrt2)`

Explanation:

To write this complex number in standard form, what is needed is value of `cos(pi/12)=cos15^@` and `sin(pi/12)=sin15^@`

`cos15^@=cos(45^@-30^@)=cos45^@cos30^@+sin45^@sin30^@`

= `1/sqrt2xxsqrt3/2+1/sqrt2xx1/2=(sqrt3+1)/(2sqrt2)`

Similarly `sin15^@=sin(45^@-30^@)=sin45^@cos30^@-cos45^@sin30^@`

= `1/sqrt2xxsqrt3/2-1/sqrt2xx1/2=(sqrt3-1)/(2sqrt2)`

Hence, `8(cos(pi/12)+isin(pi/12))`

= `8(sqrt3+1)/(2sqrt2)+i8xx(sqrt3-1)/(2sqrt2)`

= `2sqrt2(sqrt3+1)+i2sqrt2(sqrt3-1)`

= `(2sqrt6+2sqrt2)+i(2sqrt6-2sqrt2)`