 Hello and welcome to the session. In this session we will discuss a question which says that write 1 plus root 3 iota in polar form and then multiply it by 2 into cos of 5 by 6 plus iota sign 5 by 6. The whole illustrates using eigen diagram. Now before starting the solution of this question we should know some results. First is the polar form of a complex number z is equal to x plus y iota is given by z is equal to r into cos theta plus iota sign theta the whole where r is modulus of the complex number and theta is argument of the complex number and modulus of z is equal to r which is equal to square root of x square plus y square and theta that is the argument is equal to tan inverse of y upon x and second result is if z1 is equal to r1 into cos alpha plus iota sign alpha the whole and z2 is equal to r2 into cos beta plus iota sign beta the whole between complex numbers then your product z1 z2 is equal to r1 r2 into cos of alpha plus beta the whole plus iota sign of alpha plus beta whole and this complete whole. Now these results will work out as a key idea for solving out the given question. Now let us start with the solution of the given question. Now in this question we have given a complex number 1 plus root 3 iota and we have to write it in polar form so let this complex number be z and we have given z is equal to 1 plus root 3 iota. Now using the result which is given in the key idea we know the polar form of a complex number. Now we have to write this in polar form z is equal to r into cos theta plus iota sign theta the whole. First we find r now from the key idea we know that r is equal to square root of x square plus y square and is equal to modulus of z which is equal to square root of x square plus y square. Now here x is equal to 1 and y is equal to root 3 so this is equal to square root of 1 square plus root 3 whole square which is equal to square root of 1 plus 3 which is equal to root 4 that is equal to 2 so modulus of complex number z that is r is equal to 2. Now we will find argument theta that is equal to tan inverse of y upon x that is equal to tan inverse of now y is root 3 upon x is equal to tan inverse of root 3. Now we know that tan pi by 3 is root 3 so this is equal to tan inverse of tan pi by 3 which is equal to pi by 3 therefore argument theta is equal to pi by 3 now we have r is equal to 2 is equal to pi by 3 thus polar form of complex number 1 plus root 3 iota is equal to that is 2 into cos theta that is cos pi by 3 plus iota sign pi by 3 that is iota sign pi by 3 the whole. Now we have to multiply the polar form of this complex number by the given complex number in polar form like the given complex number d w which is equal to 2 into cos pi by 6 plus iota sign pi by 6 the whole whose modulus is equal to 2 and argument is equal to pi by 6 so we have the product z w is equal to 2 into cos plus iota sign pi by 3 the whole into 2 into cos pi by 6 plus iota sign pi by 6 the whole now we will use the second result which is given to us in the key idea and this is equal to 2 into 2 into cos pi by 3 plus pi by 6 the whole plus iota sign pi by 3 plus pi by 6 the whole and this complete whole now simply find we have 4 into cos of 2 pi plus pi whole pi by 6 the whole plus iota sign of 2 pi plus pi whole pi by 6 the whole and this complete whole and this is equal to 4 into cos of 3 pi by 6 plus iota sign of 3 pi by 6 the whole which is equal to 4 into cos of pi by 2 plus iota sign of pi by 2 the whole now we know that cos pi by 2 is 0 and sign pi by 2 is 1 so this is equal to 4 into 0 plus iota into 1 the whole which is equal to 4 into iota that is equal to 4 iota thus product z w is equal to 4 iota now that is represent complex numbers z w and z w on the argon plane now this is the complex number z in polar form so its polar coordinates will be given by r theta that is 2 pi by 3 all we can write it as 2 60 degrees so this is the point p with polar coordinates 2 60 degrees all we can write p is the point with rectangular coordinates 1 rule 3 that represents the complex number z which is equal to 1 plus root 3 iota now for complex number w the polar coordinates will be 2 pi by 6 all we can write it as 2 30 degrees so the point q represents the complex number w now this is the polar form of the complex number z w so its polar coordinates will be given by 4 pi by 2 all we can write it as 4 90 degrees so this is the point r with polar coordinates 4 90 degrees and this point represents the complex number z w thus we see that the product z w is again a complex number whose amplitude or argument is sum of the arguments of the two numbers that is 30 degrees plus 60 degrees which is equal to 90 degrees and modulus is the product of moduli of the two complex numbers that is 2 into 2 is equal to 4 so this is the solution of the given question that's all for this session hope you all have enjoyed the session