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

Make the numbers polar by converting them.

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

Thus, our concluding response is:

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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°)).

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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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