To express the complex number 6 + 6i in trigonometric form, we first need to find its modulus and argument.
1. **Modulus**: The modulus of a complex number a + bi is given by the formula: r = √(a² + b²). For our complex number, a = 6 and b = 6.
So, we calculate:
r = √(6² + 6²) = √(36 + 36) = √72 = 6√2.
2. **Argument**: The argument θ (theta) can be found using tan(θ) = b/a. Thus:
tan(θ) = 6/6 = 1.
This implies that θ = π/4 (since both a and b are positive, the angle lies in the first quadrant).
3. **Trigonometric Form**: The trigonometric form of a complex number is expressed as:
r(cos(θ) + i sin(θ)).
Now, substituting our values:
6√2 (cos(π/4) + i sin(π/4)).
Hence, the complex number 6 + 6i in trigonometric form is:
6√2 (cos(π/4) + i sin(π/4)).