Question

Write z equals negative 6 plus 6 i in polar form.

(1 point)
Responses

z equals 6 Start Root 2 End Root left-parenthesis cosine Start Fraction pi over 4 End Fraction plus i sine Start Fraction pi over 4 End Fraction right-parenthesis
Image with alt text: z equals 6 Start Root 2 End Root left-parenthesis cosine Start Fraction pi over 4 End Fraction plus i sine Start Fraction pi over 4 End Fraction right-parenthesis

z equals 2 Start Root 6 End Root left-parenthesis cosine Start Fraction 3 pi over 4 End Fraction plus i sine Start Fraction 3 pi over 4 End Fraction right-parenthesis
Image with alt text: z equals 2 Start Root 6 End Root left-parenthesis cosine Start Fraction 3 pi over 4 End Fraction plus i sine Start Fraction 3 pi over 4 End Fraction right-parenthesis

z equals 6 Start Root 2 End Root left-parenthesis cosine Start Fraction 3 pi over 4 End Fraction plus i sine Start Fraction 3 pi over 4 End Fraction right-parenthesis
Image with alt text: z equals 6 Start Root 2 End Root left-parenthesis cosine Start Fraction 3 pi over 4 End Fraction plus i sine Start Fraction 3 pi over 4 End Fraction right-parenthesis

z equals 6 Start Root 2 End Root left-parenthesis cosine Start Fraction pi over 4 End Fraction plus i sine Start Fraction 3 pi over 4 End Fraction right-parenthesis

Answer 1

To convert the complex number \( z = -6 + 6i \) into polar form, we first need to calculate its magnitude (modulus) and argument (angle).

1. **Magnitude**:
\[
|z| = \sqrt{(-6)^2 + (6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
\]

2. **Argument**:
The argument (θ) can be found using the arctangent function:
\[
\theta = \tan^{-1}\left(\frac{y}{x}\right) = \tan^{-1}\left(\frac{6}{-6}\right) = \tan^{-1}(-1)
\]
Since the complex number is in the second quadrant (where x is negative and y is positive), we need to add \(\pi\):
\[
\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}
\]

3. **Polar Form**:
Now we can express \( z \) in polar form:
\[
z = |z| \left(\cos(\theta) + i\sin(\theta)\right) = 6\sqrt{2} \left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right)
\]

Thus, the polar form of \( z = -6 + 6i \) is:
\[
z = 6\sqrt{2} \left(\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right)
\]

The correct response is:
**z equals 6 Start Root 2 End Root left-parenthesis cosine Start Fraction 3 pi over 4 End Fraction plus i sine Start Fraction 3 pi over 4 End Fraction right-parenthesis.**