How do you express the complex number 3 + 3i in trigonometric form?

To express a complex number in trigonometric form, we typically use the format:

z = r (cos θ + i sin θ)

where:

• r is the modulus (magnitude) of the complex number
• θ is the argument (angle) of the complex number

Let’s break down the steps to express the complex number 3 + 3i in trigonometric form:

Step 1: Calculate the Modulus (r)

The modulus of a complex number z = a + bi is calculated as:

r = √(a² + b²)

For our complex number, a = 3 and b = 3. Therefore,

r = √(3² + 3²) = √(9 + 9) = √18 = 3√2

Step 2: Calculate the Argument (θ)

The argument of a complex number can be calculated using the arctangent function:

θ = tan⁻¹(b/a)

Using our values:

θ = tan⁻¹(3/3) = tan⁻¹(1) = π/4 radians

Step 3: Substitute r and θ into the trigonometric form

Now that we have both the modulus and the argument, we can express the number in trigonometric form:

z = 3√2 (cos(π/4) + i sin(π/4))

This is the trigonometric form of the complex number 3 + 3i.

Final Result:

The final answer is:

z = 3√2 (cos(π/4) + i sin(π/4))

And that’s how you express the complex number 3 + 3i in trigonometric form!