Final answer:
To rewrite z = 2 − 8i in trigonometric form, we use the formulas for magnitude and argument. The magnitude is 2sqrt(17) and the argument is -75.96°. Therefore, z ≈ 2sqrt(17) * cis(-75.96°).
Step-by-step explanation:
To rewrite z = 2 − 8i in trigonometric form, we need to find the magnitude and argument of the complex number. The magnitude, denoted by r, can be found using the formula r = sqrt(a^2 + b^2), where a and b are the real and imaginary parts of the complex number, respectively. In this case, a = 2 and b = -8. So, r = sqrt(2^2 + (-8)^2) = sqrt(4 + 64) = sqrt(68) = 2sqrt(17).
The argument, denoted by θ, can be found using the formula tan(θ) = b/a. In this case, b = -8 and a = 2. So, tan(θ) = -8/2 = -4. Taking the arctan of -4, we get θ ≈ -75.96°.
Therefore, the trigonometric form of z = 2 - 8i is z ≈ 2sqrt(17) * cis(-75.96°), where cis(θ) = cos(θ) + i * sin(θ).