How do you divide # (-4+5i)/(6+i) # in trigonometric form?

Answer 1

#color(purple)(=> 1.05 (-0.482 + i 0.876)#

#z_1 / z_2 = r_1/ r_2 (cos (theta_1 + thet_2) + i sin (theta_1 + theta_2)#
#z_1 = (-4 + i5), z_2 = (6 + i)#
#=> r_1 = sqrt(4^2 + 5^2) = sqrt41 = 6.4#
#theta_1 = tan^(-1) (5/-4)= -51.34^@ = 128.26^@, " II Quadrant"#
#=> r_2 = sqrt(1^2 + 6^2) = sqrt37 = 6.08#
#theta_2 = tan^(-1) (1/6)= 9.46^@ #
#z_1 / z_2 = sqrt41/sqrt37 (cos (128.26 - 9.46) + i sin(128.26 - 9.46)0#
#color(purple)(=> 1.05 (-0.482 + i 0.876)#
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Answer 2

To divide complex numbers in trigonometric form, follow these steps:

  1. Convert both the numerator and denominator to polar form.
  2. Divide the magnitudes and subtract the angles.
  3. Convert the result back to rectangular form if necessary.

Given ((-4+5i)) and ((6+i)):

  1. Convert to polar form:

For ((-4+5i)): Magnitude (r_1 = \sqrt{(-4)^2 + 5^2} = \sqrt{16+25} = \sqrt{41}), Angle (\theta_1 = \arctan\left(\frac{5}{-4}\right) = \arctan(-1.25) = -0.8961) radians.

For ((6+i)): Magnitude (r_2 = \sqrt{6^2 + 1^2} = \sqrt{37}), Angle (\theta_2 = \arctan\left(\frac{1}{6}\right) = \arctan(0.1667) = 0.1594) radians.

  1. Divide the magnitudes and subtract the angles:

(r = \frac{r_1}{r_2} = \frac{\sqrt{41}}{\sqrt{37}}), (\theta = \theta_1 - \theta_2 = -0.8961 - 0.1594).

  1. Convert back to rectangular form:

(\frac{r_1}{r_2}) is the magnitude and (\theta_1 - \theta_2) is the angle in polar form.

So, the division in trigonometric form is: [\frac{-4+5i}{6+i} = \frac{\sqrt{41}}{\sqrt{37}} , \text{cis}(-1.0555 , \text{radians})]

Where "cis" represents cosine plus sine, denoting the complex number in trigonometric form.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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