# 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

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.