To determine the product of the given complex numbers [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex], we will use the properties of multiplication in trigonometric form. Specifically, when multiplying two complex numbers in trigonometric form:
[tex]\[ z = r(\cos \theta + i \sin \theta) \][/tex]
the magnitudes (radii) are multiplied and the angles (arguments) are added.
Given [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex]:
[tex]\[
z_1 = 5 \left( \cos 25^\circ + i \sin 25^\circ \right)
\][/tex]
[tex]\[
z_2 = 2 \left( \cos 80^\circ + i \sin 80^\circ \right)
\][/tex]
Step-by-step solution:
1. Magnitudes:
- Magnitude of [tex]\( z_1 \)[/tex] is [tex]\( r_1 = 5 \)[/tex]
- Magnitude of [tex]\( z_2 \)[/tex] is [tex]\( r_2 = 2 \)[/tex]
When we multiply the magnitudes:
[tex]\[ r_{\text{product}} = r_1 \times r_2 = 5 \times 2 = 10 \][/tex]
2. Angles:
- Angle of [tex]\( z_1 \)[/tex] is [tex]\( \theta_1 = 25^\circ \)[/tex]
- Angle of [tex]\( z_2 \)[/tex] is [tex]\( \theta_2 = 80^\circ \)[/tex]
When we add the angles:
[tex]\[ \theta_{\text{product}} = \theta_1 + \theta_2 = 25^\circ + 80^\circ = 105^\circ \][/tex]
Therefore, the product of the complex numbers [tex]\( z_1 \)[/tex] and [tex]\( z_2 \)[/tex] is:
[tex]\[ z_{\text{product}} = 10 \left( \cos 105^\circ + i \sin 105^\circ \right) \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{B: z_1=10\left(\cos \left(105^{\circ}\right)+i \sin \left(105^{\circ}\right)\right)} \][/tex]
