Convert Complex Numbers to Trigonometric Form

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

Formulas

Formulas:

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

Example 1

Example 1:

Here we have our number in standard form, z = a + b * i. We must convert it to the form z = r * (cosθ + i * sinθ). We know r = sqrt( a² + b²) and tanθ = b/a.

If we use the a and b of our complex number to form a triangle. The modulus of z or r or the distance between the origin and our point is the hypotenuse. Now we can use a and b as our legs to find our missing side length.

Now we know c, the hypotenuse, or our r, 2 * sqrt(5). Now, we must solve for our argument or theta, by using tanθ = b/a. Remember b is the opposite side length while a is the adjacent side length.

Since we are in Quadrant I, we have the theta that we want. We also have our r, so now we can use z = r * (cosθ + i * sinθ). Our final answer is z = 2sqrt(5) * (cos63.1° + i * sin63.1°).

Example 2

Example 2:

Here we have our number in standard form, z = a + b * i. We must convert it to the form z = r * (cosθ + i * sinθ). We know r = sqrt( a² + b²) and tanθ = b/a.

If we use the a and b of our complex number to form a triangle. The modulus of z or r or the distance between the origin and our point is the hypotenuse. Now we can use a and b as our legs to find our missing side length. But before we graph we have simplify the number we have in the question, 2i * (3 + 2i).

Here is where trouble starts. The calculator doesn't know which quadrant the triangle is in. This means it doesn't know if a or b is negative. We, the humans, know, so we have to use our drawing to figure out what the calculator is telling us. If we go counterclockwise, the negative direction, Quadrant IV is from 0° to -90°. That means the -56.3° is in Quadrant IV, and gives us a different triangle than our original. To find θ₁= we need to add 180° to -56.3°, which is 123.7°.

If we use positive numbers for both a and b, we know the angle is 56.3°.

Since θ₁ and θ₂ form a linear pair, the we can subtract θ₂ from 180° to find θ₁. So 180° minus 56.7° is 123.7°.

We cannot forget about r. Let's solve for that as well.

We have the theta that we want. We also have our r, so now we can use z = r * (cosθ + i * sinθ). Our final answer is z = 2sqrt(13) * (cos127.3° + i * sin127.3°).

Example 3

Example 3:

Here we have our number in standard form, z = a + b * i. We must convert it to the form z = r * (cosθ + i * sinθ). We know r = sqrt( a² + b²) and tanθ = b/a.

If we use the a and b of our complex number to form a triangle. The modulus of z or r or the distance between the origin and our point is the hypotenuse. Now we can use a and b as our legs to find our missing side length.

Now we know c, the hypotenuse, or our r, 4. Now, we must solve for our argument or theta, by using tanθ = b/a. Remember b is the opposite side length while a is the adjacent side length.

Since θ₁ and θ₂ form a complete circle pair, the we can add 2π and θ₂ to find θ₁. So 2π plus -π/6 is 11π/6.

We have the theta that we want. We also have our r, so now we can use z = r * (cosθ + i * sinθ). Our final answer is z = 4 * [cos(11π/6) + i * sin(11π/6)].

Live Worksheet

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