Answer:
[tex]10(\cos 75^\circ+\mathbf{i}\sin 75^\circ)[/tex]
Step-by-step explanation:
Complex Numbers
Complex numbers can be expressed in several forms. One of them is the rectangular form(x,y):
[tex]Z = x+\mathbf{i}y[/tex]
Where
[tex]\mathbf{i}=\sqrt{-1}[/tex]
They can also be expressed in polar form (r,θ):
[tex]Z=r(\cos\theta+\mathbf{i}\sin\theta)[/tex]
The polar form is also shortened to:
[tex]Z = r CiS(\theta)[/tex]
The product of two complex numbers in polar form is:
[tex][r_1Cis(\theta_1)]\cdot [r_2Cis(\theta_2)]=r_1\cdot r_2Cis(\theta_1+\theta_2)[/tex]
We are given the complex numbers:
2(cos(45°) + i sin(45°)) and 5(cos(30°) + i sin(30°))
They can be written as:
2CiS(45°) and 5CiS(30°). The product is:
2CiS(45°) * 5CiS(30°) = 10CiS(75°)
Expressing back in rectangular form:
[tex]\boxed{2CiS(45^\circ) \cdot 5CiS(30^\circ) =10(\cos 75^\circ+\mathbf{i}\sin 75^\circ)}[/tex]
