Argument of a complex number in different quadrants :
Let (r, θ) be the polar co-ordinates of the point.
P = P(x, y) in the complex plane corresponding to the complex number
z = x + iy
cos θ = Adjacent side/hypotenuse side ==> OM/MP ==> x/r
sin θ = Opposite side/hypotenuse side ==> PM/OP ==> y/r
x = r cos θ and y = r sin θ
tan θ = y/x
θ = tan−1 y/x
is called the amplitude or argument of z = x + iy
denoted by amp z or arg z and is measured as the angle which the line OP makes with the positive x-axis (in the anti clockwise sense).
Usually we have two methods to find the argument of a complex number
(i) Using the formula θ = tan−1 y/x
here x and y are real and imaginary part of the complex number respectively.
This formula is applicable only if x and y are positive.
But the following method is used to find the argument of any complex number.
To find the modulus and argument for any complex number we have to equate them to the polar form
r (cos θ + i sin θ)
Here r stands for modulus and θ stands for argument
Let us see some example problems to understand how to find the modulus and argument of a complex number.
Example 1 :
Find the modulus and argument of the complex number
- √2 + i √2
Solution :
- √2 + i √2 = r (cos θ + i sin θ) ----(1)
r = √ [(-√2)² + √2²] = √(2 + 2) = √4 = 2
r = 2
Apply the value of r in the first equation
- √2 + i √2 = 2 (cos θ + i sin θ)
- √2 + i √2 = 2 cos θ + i 2 sin θ
Equating the real and imaginary parts separately
2 cos θ = - √2 cos θ = - √2/2 cos θ = - 1/√2 |
2 sin θ = √2 sin θ = √2/2 sin θ = 1/√2 |
Since sin θ is positive and cos θ is negative the required and θ lies in the second quadrant.
θ = Π - α
Here α is nothing but the angles of sin and cos for which we get the value 1/√2
θ = Π - (Π/4)
θ = (4Π-Π)/4 ==> 3Π/4
Modulus = 2 and argument = 3Π/4
Hence - √2 + i √2 = 2 (cos 3Π/4 + i sin 3Π/4)
Example 2 :
Find the modulus and argument of a complex number
1 + i √3
Solution :
1 + i √3 = r (cos θ + i sin θ) ----(1)
r = √ [(1)² + √3²] = √(1 + 3) = √4 = 2
r = 2
Apply the value of r in the first equation
1 + i √3 = 2 (cos θ + i sin θ)
1 + i √3 = 2 cos θ + i 2 sin θ
Equating the real and imaginary parts separately
2 cos θ = 1 cos θ = 1/2 |
2 sin θ = √3 sin θ = √3/2 |
Since sin θ and cos θ are positive, the required and θ lies in the first quadrant.
θ = α
Here α is nothing but the angles of sin and cos for which we get the values 1/2 and √3/2 respectively.
θ = Π/3
Modulus = 2 and argument = Π/3
Hence - √2 + i √2 = 2 (cos Π/3 + i sin Π/3)
Example 3 :
Find the modulus and argument of the complex number
-1 - i √3
Solution :
-1 - i √3 = r (cos θ + i sin θ) ----(1)
r = √ [(-1)² + (-√3)²] = √(1 + 3) = √4 = 2
r = 2
Apply the value of r in the first equation
-1 - i √3 = 2 (cos θ + i sin θ)
-1 - i √3 = 2 cos θ + i 2 sin θ
Equating the real and imaginary parts separately
2 cos θ = -1 cos θ = -1/2 |
2 sin θ = -√3 sin θ = -√3/2 |
Since sin θ and cos θ are negative the required and θ lies in the third quadrant.
θ = -Π + α
Here α is nothing but the angles of sin and cos for which we get the values √3/2 and 1/2 respectively.
θ = - Π + α
= - Π + Π/3 ==> (-3Π+Π)/3 ==>-2Π/3
Modulus = 2 and argument =-2Π/3
Hence - 1 - i √3 = 2 (cos (-2Π/3) + i sin (-2Π/3))
After having gone through the stuff given above, we hope that the students would have understood "Argument of a complex number in different quadrants".
Apart from the stuff given above, if you want to know more about "argument of a complex number in different quadrants", please click here
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits