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Argand Plane & Polar Representation Of Complex Number, Argand Plane, We all know that the pair of numbers (x, y) can be represented on the XY-plane, where x is, called abscissa and y is called the ordinate. Similarly, we can represent complex numbers also on, a plane called Argand plane or complex plane. Similar to the X-axis and Y-axis in twodimensional geometry, there are two axes in Argand plane., The plane having a complex number assigned to each of its point is called the complex, plane or the Argand plane., , , The axis which is horizontal is called the real axis, The axis which is vertical is called the imaginary axis, The complex number x+iy which corresponds to the ordered pair(x, y)is represented, geometrically as the unique point (x, y) in the XY-plane., , For example,, The complex number, 2+3i corresponds to the ordered pair (2, 3) geometrically., Similarly, -3+2i corresponds to the ordered pair (-3, 2)., , 1
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, , , Complex numbers in the form 0+ai, where “a” is any real number will lie on the, imaginary axis., Complex numbers in the form a+0i, where “a” is any real number will lie on the real axis., , It is obvious that the modulus of complex number x+iy ,, origin (0, 0) and the point p (x, y)., , , is the distance between the, , The conjugate of z = x+iy is z = x-iy which is represented as (x, -y) in the Argand plane., Point (x, -y) is the mirror image of the point (x, y) across the real axis in the Argand, plane., , Example: Find the distance between the complex number z = 3 – 4i and the origin in the Argand, plane?, We know that,, Distance between the origin and given complex no. z= 3 – 4i is equal to the modulus of z., |z|=, , =, , =, , = 5 units, , Polar representation of complex numbers:, POLAR FORM :, INTRODUCTION:, Z=x +iy, , rectangular co-ordinate form., , And z = r (cosθ + i sinθ), = amplitude ( argument), , polar form where r= modulus =, i.e. tan, Or, , =, = tan - 1, , Angles in terms of :, , 180, , 90, , 60, , =, , =, , =, , 2, , 45, =, , 30, =, , 360, =2, , and
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Let “P” represent the non-zero complex number x + iy. OP is the directed line segment of length, r and makes an angle θ with the positive direction of X-axis. Ordered pair (r, θ) is called as the, polar coordinates of the point P, as the point “P” is uniquely determined by (r, θ). The origin is, called the pole and the positive X-axis is called the initial line., Then,, x = r cosθ, y = r sinθ, We can write ,, z = x + iy, As z = r cosθ + ir sinθ = r (cosθ + i sinθ),, number., , which is called the polar form of complex, , Here, r = |z| =, is modulus of z and θ is known as the argument or amplitude of z, denoted as arg z, For any non-zero complex number z, there corresponds one value of θ, in the interval, [0, 2π), In any other interval of length 2π, for example, consider the interval -π < θ ≤ π, then the, value of θ is called the principal argument of z., Example: Represent z =, + i in the polar form, , , Here,, x=, , y= 1, r = |z| =, Now, tan θ =, or tan θ =, , 6, , =, , =, , =, , =2
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or tan = tan 30, , [ since tan, , θ=, , [ since, , 30 ], = 30 and, , x is +ve and y is +ve so, , Therefore, polar form of z is,, z ={ 2 (cos ( ) + i sin ( )}, , Q1, Soln : Here,, x=, , y=, Modulus = |z| =, , =, , =, , =, , =2, , Now, tan θ =, or tan θ =, or tan, or, , = tan 60, , =, [ since tan 60 =, , ], , = 60, , θ = - [as x is –ve and y is –ve so, =, =, Q2. Soln : Here,, , 7, , [since, , = 60 ], , lies in 3rd quadrant], , lies in first quadrant]
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x=, , y=, Modulus = |z| =, , =, , =, , =, , =2, , Now, tan θ =, or tan θ =, or tan, or, , =, , = tan 30, , [ since tan 30 =, , ], , = 30, , θ= -, , lies in 2rd quadrant], , [as x is -ve and y is +ve so, , =, , [since, , = 30 ], , =, Q3. Represent z = 1- i in the polar form, Soln :, , Here,, x= 1, , y=- 1, r = |z| =, , =, , =, , =, , Now, tan θ =, or tan θ =, , = -1, , or tan = tan 45, , [ since tan 45, , =1], , or, θ=-, , [ as x is +ve and y is -ve so, , =[ since, The polar form of z is, Z = r (cos + isin ), z=, (cos ( ) + i sin (, , lies in 4th quadrant], , = 45 ], , ), , Q4. Represent z = -1+ i in the polar form, Soln :, , Here,, x= -1, , y=, r = |z| =, , 8, , =, , 1, =, , =
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Now, tan θ =, or tan θ =, , = -1, , or tan = tan 45, , [ since tan 45, , =1], , or, θ=, , -, , =, , -, , [ since, , lies in 2th quadrant], , [ since, , =, The polar form of z is, Z = r (cos + isin ), z=, (cos (, ) + i sin (, , = 45 ], , ), , Q5. Represent z = -1- i in the polar form, Soln :, , Here,, x= - 1, , y= - 1, r = |z| =, , =, , =, , =, , Now, tan θ =, or tan θ =, , =1, , or tan = tan 45, , [ since tan 45, , =1], , or, θ=, , -, , =, , -, , [as x is -ve and y is -ve so, [ since, , = 45 ], , =, The polar form of z is, Z = r (cos + isin ), z=, (cos (, ) + i sin (, ), Q6. Represent z = -3 in the polar form, Soln :, , Here,, z=-3, z = - 3 + i0, x= -3, , y= 0, r = |z| =, Now, tan θ =, , 9, , =, , =, , =3, , lies in 3th quadrant]
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or tan θ =, , =0, , or tan =, , [ since, , lies on x axis as x is –ve and y=0 so tan, , or, or, The polar form of z is, Z = r (cos + isin ), z=, (cos + i sin ), Example:, , Represent z =, , + i in the polar form, , Here,, x=, , y= 1, r = |z| =, , =, , =, , =, , =2, , Now, tan θ =, or tan θ =, or tan = tan 30, θ=, , [since, , [ since tan, = 30 and, , The polar form of z is,, z ={ 2 (cos ( ) + i sin ( )}, , HOME WORK; Q8, , 10, , 30 ], lies in first quadrant], , =0 ]
