The polar type of a complicated number is another method to stand for a complicated number. The kind z = a + b ns is dubbed the rectangular coordinate type of a complex number.
The horizontal axis is the actual axis and also the vertical axis is the imaginary axis. We discover the actual and complicated components in regards to r and also θ where r is the length of the vector and θ is the edge made with the actual axis.
indigenous Pythagorean organize :
r 2 = a 2 + b 2
By using the simple trigonometric ratios :
cos θ = a r and also sin θ = b r .
Multiplying every side by r :
r cos θ = a and also r sin θ = b
The rectangular form of a facility number is given by
z = a + b i .
instead of the worths of a and b .
z = a + b i = r cos θ + ( r sin θ ) ns = r ( cos θ + ns sin θ )
In the situation of a complex number, r represents the absolute value or modulus and the edge θ is referred to as the debate of the complicated number.
This have the right to be summarized as follows:
The polar type of a complex number z = a + b i is z = r ( cos θ + i sin θ ) , where r = | z | = a 2 + b 2 , a = r cos θ and b = r sin θ , and also θ = tan − 1 ( b a ) for a > 0 and θ = tan − 1 ( b a ) + π or θ = tan − 1 ( b a ) + 180 ° for a 0 .
You are watching: When a complex number z is written in polar form
refer the complex number in polar form.
5 + 2 ns
The polar kind of a complicated number z = a + b i is z = r ( cos θ + ns sin θ ) .
So, an initial find the absolute worth of r .
r = | z | = a 2 + b 2 = 5 2 + 2 2 = 25 + 4 = 29 ≈ 5.39
Now find the debate θ .
because a > 0 , usage the formula θ = tan − 1 ( b a ) .
θ = tan − 1 ( 2 5 ) ≈ 0.38
note that right here θ is measure up in radians.
See more: Guide To Tcp/Ip 4Th Edition Pdf, Download Guide To Tcp/Ip, 4Th Edition Pdf
Therefore, the polar type of 5 + 2 ns is around 5.39 ( cos ( 0.38 ) + ns sin ( 0.38 ) ) .