Answer:
z² = 4(cos(208°) +i·sin(208°))
z² = 4·cos(208°) +i·4·sin(208°) ≈ -3.5318 -1.8779i
Explanation:
You want the polar and rectangular forms of the square of z=2(cos(104°)+i·sin(104°)).
Polar form
The number is given in polar form. We can square that to find the desired value.
z = 2(cos(104°) +i·sin(104°))
z² = 2²(cos(104°)² +2·i·cos(104°)sin(104°) +i²·sin(104°)²)
= 4(cos(104°)² -sin(104°)² +i·2·sin(104°)cos(104°))
Using the trig identities ...
- cos(2θ) = cos(θ)² -sin(θ)²
- sin(2θ) = 2sin(θ)cos(θ)
we have ...
z² = 4(cos(208°) +i·sin(208°))
Rectangular form
The rectangular form is the expansion of the above polar form:
z² = 4·cos(208°) +i·4·sin(208°) ≈ -3.5318 -1.8779i
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Additional comment
In general, a power of a complex number in polar form will be ...
(a∠θ)^n = (a^n)∠(n·θ)
(2∠104°)² = 2²∠(2·104°) = 4∠208° . . . . as shown above
A number of different notations are used to express complex numbers in polar form. Effectively, you can choose from any of ...
r(cos(θ)+i·sin(θ)) ⇔ r∠θ ⇔ (r; θ) ⇔ r·e^(i·θ) ⇔ e^(ln(r)+i·θ)
The angle value in the exponential forms will be in radians.
The exponential forms help you see that products and ratios are easily found:
(a∠α)(b∠β) = ab∠(α+β)
(a∠α)/(b∠β) = (a/b)∠(α-β)