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 .

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Example:

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.

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Therefore, the polar type of 5 + 2 ns is around 5.39 ( cos ( 0.38 ) + ns sin ( 0.38 ) ) .