At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
To determine if the complex number [tex]\(-1 + i\sqrt{3}\)[/tex] is equal to [tex]\(2 \text{ cis } 120^\circ\)[/tex] in polar form, we need to verify both the magnitude (or modulus) and the angle (or argument) of the complex number in polar form.
Step 1: Compute the Magnitude
The magnitude [tex]\(r\)[/tex] of a complex number [tex]\(a + bi\)[/tex] is given by the formula:
[tex]\[ r = \sqrt{a^2 + b^2} \][/tex]
For the complex number [tex]\(-1 + i\sqrt{3}\)[/tex]:
- The real part [tex]\(a\)[/tex] is [tex]\(-1\)[/tex].
- The imaginary part [tex]\(b\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Let's compute the magnitude:
[tex]\[ r = \sqrt{(-1)^2 + (\sqrt{3})^2} \][/tex]
[tex]\[ r = \sqrt{1 + 3} \][/tex]
[tex]\[ r = \sqrt{4} \][/tex]
[tex]\[ r = 2 \][/tex]
So, the magnitude of [tex]\(-1 + i\sqrt{3}\)[/tex] is [tex]\(2\)[/tex].
Step 2: Compute the Angle
The angle [tex]\(\theta\)[/tex] of a complex number [tex]\(a + bi\)[/tex] in polar form is given by:
[tex]\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \][/tex]
Note: The angle should be adjusted based on the quadrant in which the complex number lies.
For the complex number [tex]\(-1 + i\sqrt{3}\)[/tex]:
[tex]\[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) \][/tex]
This evaluation gives an angle in the second quadrant because the real part is negative and the imaginary part is positive. The exact angle is:
[tex]\[ \theta = 180^\circ - \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) \][/tex]
[tex]\[ \theta = 180^\circ - 60^\circ \][/tex]
[tex]\[ \theta = 120^\circ \][/tex]
So, the angle of [tex]\(-1 + i\sqrt{3}\)[/tex] in polar form is [tex]\(120^\circ\)[/tex].
Conclusion
- The magnitude of [tex]\(-1 + i\sqrt{3}\)[/tex] is [tex]\(2\)[/tex].
- The angle of [tex]\(-1 + i\sqrt{3}\)[/tex] is [tex]\(120^\circ\)[/tex].
Therefore, the complex number [tex]\(-1 + i\sqrt{3}\)[/tex] indeed corresponds to [tex]\(2 \text{ cis } 120^\circ\)[/tex] in polar form.
The statement [tex]\(-1 + i\sqrt{3} = 2 \text{ cis } 120^\circ\)[/tex] is True.
Step 1: Compute the Magnitude
The magnitude [tex]\(r\)[/tex] of a complex number [tex]\(a + bi\)[/tex] is given by the formula:
[tex]\[ r = \sqrt{a^2 + b^2} \][/tex]
For the complex number [tex]\(-1 + i\sqrt{3}\)[/tex]:
- The real part [tex]\(a\)[/tex] is [tex]\(-1\)[/tex].
- The imaginary part [tex]\(b\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Let's compute the magnitude:
[tex]\[ r = \sqrt{(-1)^2 + (\sqrt{3})^2} \][/tex]
[tex]\[ r = \sqrt{1 + 3} \][/tex]
[tex]\[ r = \sqrt{4} \][/tex]
[tex]\[ r = 2 \][/tex]
So, the magnitude of [tex]\(-1 + i\sqrt{3}\)[/tex] is [tex]\(2\)[/tex].
Step 2: Compute the Angle
The angle [tex]\(\theta\)[/tex] of a complex number [tex]\(a + bi\)[/tex] in polar form is given by:
[tex]\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \][/tex]
Note: The angle should be adjusted based on the quadrant in which the complex number lies.
For the complex number [tex]\(-1 + i\sqrt{3}\)[/tex]:
[tex]\[ \theta = \tan^{-1}\left(\frac{\sqrt{3}}{-1}\right) \][/tex]
This evaluation gives an angle in the second quadrant because the real part is negative and the imaginary part is positive. The exact angle is:
[tex]\[ \theta = 180^\circ - \tan^{-1}\left(\frac{\sqrt{3}}{1}\right) \][/tex]
[tex]\[ \theta = 180^\circ - 60^\circ \][/tex]
[tex]\[ \theta = 120^\circ \][/tex]
So, the angle of [tex]\(-1 + i\sqrt{3}\)[/tex] in polar form is [tex]\(120^\circ\)[/tex].
Conclusion
- The magnitude of [tex]\(-1 + i\sqrt{3}\)[/tex] is [tex]\(2\)[/tex].
- The angle of [tex]\(-1 + i\sqrt{3}\)[/tex] is [tex]\(120^\circ\)[/tex].
Therefore, the complex number [tex]\(-1 + i\sqrt{3}\)[/tex] indeed corresponds to [tex]\(2 \text{ cis } 120^\circ\)[/tex] in polar form.
The statement [tex]\(-1 + i\sqrt{3} = 2 \text{ cis } 120^\circ\)[/tex] is True.
Thanks for using our service. We aim to provide the most accurate answers for all your queries. Visit us again for more insights. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.