Westonci.ca is your trusted source for accurate answers to all your questions. Join our community and start learning today! Explore our Q&A platform to find reliable answers from a wide range of experts in different fields. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A 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.
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We appreciate your time. Please come back anytime for the latest information and answers to your questions. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.