Convert Complex Numbers From Trigonometric to Standard Form

Convert Complex Numbers From Trigonometric to Standard Form - How it Works - Video

Formulas

Formulas:

These are the formulas that we will use to transform a complex number into standard form from trigonometric from.

Example 1

Example 1:

Here we have our number in trigonometric form, z = r * (cosθ + i * sinθ). We must convert it to the form z = a + b * i. We know a = r * cosθ and b = r * sinθ. Since θ equals π, we can substitute π into both cosine and sine.

a = 5 * cosθ

a = 5 * cos π

a = 5 * -1

a = -5


Substituted π

Evaluated cosine

Multiplied

b = 5 * sinθ

b = 5 * sin π

b = 5 * 0

b = 0


Substituted π

Evaluated sine

Multiplied

Now we go back to the standard form of a complex number, z = a + bi. We know a = -5 and b = 0 so we substitute both numbers so z = -5 + 0i. Our final answer is z = -5.

Example 2

Example 2:

Here we have our number in trigonometric form, z = r * (cosθ + i * sinθ). We must convert it to the form z = a + b * i. We know a = r * cosθ and b = r * sinθ.

In this case we don't have an exact value for θ. We have to use θ = tan-1(3). Now, we have to work backwards and draw to find the modulus of z or r or the distance from the origin to our point.

We know tangent = the opposite side length over the adjacent side length. Since θ = tan-1(3), we tan both sides to get, tanθ = 3 or tanθ = 3/1. So the opposite side length is 3 and the adjacent side length is 1. We can draw our lines on the coordinate to form a triangle.

Now, we can use the Pythagorean Theorem to find the modulus of z or r or the distance from the origin to our point.

12 + 32 = c2

1 + 9 = c2

10 = c2

c = sqrt(10)


Squared the numbers

Added

Squared root both sides

Let's go back to what we know. We know a = r * cosθ and b = r * sinθ. Now, we can use our right triangle and trigonometry to find cosθ and sinθ. Since cosine is the adjacent side length over the hypotenuse, cosθ = 1/sqrt(10). Since sine is the opposite side length over the hypotenuse, sinθ = 3/sqrt(10).

Now, we can substitute this information into both a = r * cosθ and b = r * sinθ.

a = r * cosθ

a = sqrt(10) * 1 / sqrt(10)

a = 1


Substituted sqrt(10) for r and 1/sqrt(10) for cosθ

Multiplied

b = r * sinθ

b = sqrt(10) * 3 / sqrt(10)

b = 3


Substituted sqrt(10) for r and 3/sqrt(10) for cosθ

Multiplied

Let's go back to what we know. We know z = a + b * i. a = 1 and b = 3, so let's substitute those numbers into the standard form, z = 1 + 3i is our final answer.

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