 Hello and welcome to the session. In this session, we will discuss trigonometric or polar form of a complex number and argument of a complex number. First of all, let us discuss trigonometric or polar form of a complex number. Now the polar coordinate system is two dimensional system in which each point on the plane is determined by a distance from a fixed point and an angle from a fixed direction. Now we know that a complex number is of the form, that is, let us take a complex number z which is equal to a plus b iota where a and b belong to the set of real numbers. So the complex number is of the form a plus b iota where a and b belong to the set of real numbers. Now we can represent the complex number z is equal to a plus b iota by the point b whose coordinates are av. So this is how we can represent the complex number a plus b iota graphically. Now in polar coordinate system, this point b which is representing the number that is the complex number a plus b iota. So the point b has the coordinates av. So this point b can be specified by giving a distance r of the point b from the origin that is this distance and the angle theta between the line joining the origin to the point b at the positive x axis that is this angle b theta that is the angle between the line joining the origin to the point that is to the given point which is representing the complex number which means this line r at the positive x axis. So the angle between them is called the angle theta. Now by Pythagoras theorem in this triangle, now let this be point r. So by applying Pythagoras theorem in the triangle opr we get r squared is equal to a square plus b square which implies r is equal to square root of a square plus b square. Now we know that for a complex number z which is equal to a plus b iota, modulus of z is equal to square root of a square plus b square. So here we are getting r is equal to square root of a square plus b square that means r is equal to the modulus of z. Also in the triangle opr perpendicular over base that is b over a will be equal to tan theta. So we have tan theta is equal to b over n which implies theta is equal to tan inverse of b over n. Now consider the same triangle again and in this we are getting perpendicular over hypotenuse that is b over r is equal to sin theta. So we have b over r is equal to sin theta which implies b is equal to r sin theta. Also a over r is equal to cos theta. So we are getting the coordinates as a is equal to r cos theta and b is equal to r sin theta. Now we have complex numbers z is equal to a plus b iota therefore substituting the values of a and b in this we get the polar form of the complex numbers z is now a is equal to r cos theta. So it is r cos theta plus b over r sin theta plus b over r sin theta. So we are getting iota into now b is r sin theta. So it is iota into r sin theta which is further equal to now taking r common it will be r into cos theta plus iota sin theta which can be written as r sis theta. Now in this c is representing cos i is representing the imaginary number iota as s represents sin. Now z is equal to a plus b iota is called the rectangular form and z is equal to r sis theta that is r into cos theta plus iota sin theta the whole is called the polar form or polar representation of the complex number z and the polar form is also called the trigonometric form. Now we know that in the complex number z which is equal to a plus b iota a is called the real part and b is called the imaginary part of the complex number z. So here we are getting r is equal to square root that is r is equal to root a square plus b square that means r is equal to square root of the square of the real part of z plus the square of the imaginary part of z and here we are getting theta is equal to tan inverse b over a which means theta is equal to tan inverse the imaginary part of z which is b over the real part of z which is a and r is a positive number is the absolute value or modulus of a complex number. So in the polar coordinate system the complex number a plus b iota is represented by the point r theta and we have got a is equal to r cos theta b is equal to r sin theta and r is equal to square root of a square plus b square which is equal to the modulus of z which is just the distance from the origin to the point b which is representing the complex number a plus b iota. Therefore r is a positive number and here theta is equal to tan inverse b over a. Now let us discuss some remarks first is the angle theta is measured in gradients 0 is less than equal to theta is less than 2 pi and secondly every point in the plane polar coordinates theta where r is written as equal to 0 0 is less than equal to theta is less than 2 pi or conversely you can say that for every positive value of r and each value of theta between 0 and 2 pi we get a unique point in the complex plane with polar coordinates r theta and for the point b the segment OP is called the radius vector and next r is equal to 0 if and only the complex number z is equal to 0. Now let us discuss argument of a complex number now we have already discussed the polar coordinate system now for the complex number z which is equal to a plus b iota which is represented by this point this angle that is theta is called the argument or amplitude of the complex number z which is equal to a plus b iota therefore we can write theta is equal to argument of set or amplitude of set and also we know that theta is equal to tan inverse e over n therefore an argument or amplitude of the complex number z which is equal to a plus b iota is an angle theta with initial side the positive x axis and the terminal side the ring from the origin containing the complex number a plus b iota. Now let us discuss some remarks first is the value of theta where minus pi is less than theta is less than equal to pi is called the principal value of the argument and is written as the principal value of the argument of set so by amplitude we mean the principal value and secondly if the complex number z is equal to 0 plus 0 iota then argument of set which is equal to theta which is equal to tan inverse b over a will be equal to tan inverse 0 over 0 which is not defined for a positive real number argument that is theta is equal to and for negative real number the argument theta is m d q s and is plus minus pi however we shall take it as pi now the argument or amplitude for a positive imaginary number s theta which is equal to pi by 2 and for the negative imaginary number theta is equal to minus pi by 2 and amplitude of z 1 into z 2 the whole is equal to amplitude of z 1 plus amplitude of z 2 and in general amplitude of z 1 into z 2 into z 3 and so on up to z n is equal to amplitude of z 1 plus amplitude of z 2 plus amplitude of z 3 plus so on up to amplitude of z n and amplitude of z 1 over z 2 the whole is equal to amplitude of z 1 minus amplitude of z 2 now let us discuss how to find the principal value of the argument of complex number z where z is equal to a plus b i yota so here we will discuss how to find the principal value of the argument of the complex number z line in different coordinates for this in the first step find the value of tan inverse mod of b over a line between 0 and pi by 2 and let it be alpha then in the step to find the quadrant in which the given complex number which is represented by p a b that is the point whose coordinates are a b lies now suppose we have determined the angle alpha then let us check how to find the argument of the complex number now consider these four diagrams in the first diagram if a is greater than 0 and b is greater than 0 in the complex number a plus b i yota then the point p a b which is representing the complex number will lie in the first quadrant now here this is the angle alpha now we know that the argument of a complex number a plus b i yota is an angle theta with initial size the positive x axis and terminal size the way from the origin containing a plus b i yota so here theta will be equal to that means argument of a complex number z which is equal to a plus b i yota is equal to second case if a is less than 0 and b is greater than 0 then the point p a b will lie in the second quadrant now here this is the angle alpha now we know that argument of a complex number that is the angle theta with initial size the positive x axis and the terminal size the way from the origin containing a plus b i yota that is containing the point p a b is the angle theta which is equal to y minus element of the complex number a plus b i yota is equal to down in the third case if a is less than 0 and b is less than 0 then the point p a b will lie in the third quadrant now here this is the angle alpha now by the definition of argument of the complex number that is the complex number z which is equal to a plus b i yota is the angle theta that is this is the angle theta with initial size the positive x axis and the terminal size the way from the origin containing the point p a b so theta will be equal to and also in the clockwise direction theta is equal to minus of alpha the whole of the complex number z is equal to pi plus alpha or minus of pi minus alpha the whole now in the fourth case if a is greater than 0 and b is less than 0 then in that case p a b will lie in the fourth quadrant that is the angle x op is the angle alpha so here by the definition of argument of a complex number theta in the clockwise direction will be equal to minus alpha theta that is the angle with initial size the positive x axis and terminal size the way from the original region containing the point p a b is equal to minus alpha therefore in this case the argument of the complex number z is equal to minus alpha or 2 pi minus alpha so in this session we have learnt about trigonometric or polar form of a complex number equivalent of a complex number so this completes our session hope you all have enjoyed the session