How do you multiply # (4-5i)(2+7i) # in trigonometric form?

Question

How do you multiply #  (4-5i)(2+7i) # in trigonometric form?

Daniel Acosta · Accepted Answer

#color(purple)(z_1z_2=sqrt2173[cos(0.396)+isin(0.396)])#

Make the numbers polar by converting them.

#color(red)(z_1=4-5i)#

#rArrr_1=sqrt(4^2+(-5)^2)=sqrt(16+25)=sqrt41#

#rArrtheta_1=arctan((-5)/4)~~5.387#

#rArrr_1[costheta_1+isintheta_1]=color(red)(sqrt41[cos(5.387)+isin(5.387)])#

#color(blue)(z_2=2+7i)#

#rArrr_2=sqrt(2^2+7^2)=sqrt(4+49)=sqrt53#

#rArrtheta_2=arctan(7/2)~~1.292#

#rArrr_2[costheta_2+isintheta_2]=color(blue)(sqrt53[cos(1.292)+isin(1.292)])#

When you multiply them all together now, the outcome is:

#z_1z_2=r_1r_2[cos(theta_1+theta_2)+isin(theta_1+theta_2)]#

#rArrr_1r_2=sqrt41sqrt53=sqrt(41*53)=sqrt(2173)#

#rArrtheta_1+theta_2=arctan((-5)/4)+arctan(7/2)~~6.680#

We usually try to express #theta# on the interval

#0 < theta < 2pi#

#6.680 - 2pi~~0.396#

Thus, our concluding response is:

#color(purple)(z_1z_2=sqrt2173[cos(0.396)+isin(0.396)])#

Elijah Alexander · Answer

undefined To multiply (4 - 5i)(2 + 7i) in trigonometric form, first, expand the expression using the distributive property. Then, represent each complex number in trigonometric form (polar form), which is in the form of r(cosθ + isinθ). Finally, multiply the magnitudes and add the angles.

(4 - 5i)(2 + 7i)

= 4 * 2 + 4 * 7i - 5i * 2 - 5i * 7i
= 8 + 28i - 10i - 35i^2
= 8 + 28i - 10i + 35  (since i^2 = -1)
= 43 + 18i

Now, let's represent each complex number in trigonometric form:

4 - 5i:
Magnitude (r1) = √(4^2 + (-5)^2) = √(16 + 25) = √41
Argument (θ1) = arctan(-5/4) ≈ -51.34°

2 + 7i:
Magnitude (r2) = √(2^2 + 7^2) = √(4 + 49) = √53
Argument (θ2) = arctan(7/2) ≈ 74.05°

Now, multiply the magnitudes and add the angles:

Magnitude = √41 * √53 ≈ √(41 * 53) ≈ √(2173) ≈ 46.61
Argument = -51.34° + 74.05° ≈ 22.71°

So, (4 - 5i)(2 + 7i) in trigonometric form is approximately 46.61(cos(22.71°) + isin(22.71°)).