Homework 6: Complex Numbers In Polar Form

Homework 6 Complex Numbers In Polar Form

Suppose we have a complex number expressed in polar form and we want to express it in rectangular form. (That is, we know r and θ and we need a and b.) Referring to the figure we see that we can use the formulas: Rectangular → Polar Conversion On the other hand, suppose we have a complex number expressed in rectangular form and we want to express it in polar form. (That is, we know a and b and we need r and θ .) We see that we can use the formulas: Exercise Q1. Represent 1+ i3    graphically and write it in polar form. Q2. Represent 2    − i√2    graphically and write it in polar form. Q3. Represent graphically and give the rectangular form of 6(cos 180° + i sin 180°) Q4. Represent graphically and give the rectangular form of 7.32  − 270° Q5. Convert the complex number 5 + 2 i to polar form.

Paper For Above instruction

Complex numbers are fundamental in mathematics, particularly in fields like engineering, physics, and applied mathematics. They encompass two primary forms: rectangular (or Cartesian) form and polar form. Transitioning between these two representations allows for easier computation, analysis of properties such as magnitude and phase, and visualization of complex quantities.

Rectangular and Polar Forms of Complex Numbers

The rectangular form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary part. Conversely, the polar form is characterized by the magnitude (r) and the angle (θ) in degrees or radians, denoted as r(cos θ + i sin θ).

Conversion between these forms involves fundamental formulas. Given a complex number in rectangular form with components a and b, its polar form can be computed as follows:

  • Magnitude (r): \( r = \sqrt{a^2 + b^2} \)
  • Angle (θ): \( \theta = \arctan(\frac{b}{a}) \) (adjusted for quadrant)

Conversely, for a complex number in polar form with magnitude r and angle θ, the rectangular form is obtained by:

  • Real part (a): \( a = r \cos \theta \)
  • Imaginary part (b): \( b = r \sin \theta \)

Application to Specific Complex Numbers

Q1: Represent 1 + 3i graphically and write it in polar form

Graphically, 1 + 3i is a point located one unit along the real axis and three units along the imaginary axis. Its magnitude is \( r = \sqrt{1^2 + 3^2} = \sqrt{10} \approx 3.162 \). The angle θ, in degrees, is \( \theta = \arctan(\frac{3}{1}) \approx 71.565° \). The polar form is therefore approximately \( 3.162 (cos 71.565° + i sin 71.565°) \).

Q2: Represent 2 − i√2 graphically and write it in polar form

Graphically, this point is 2 units along the real axis and -√2 units along the imaginary axis. Its magnitude is \( r = \sqrt{2^2 + (-\sqrt{2})^2} = \sqrt{4 + 2} = \sqrt{6} \approx 2.45 \). The angle θ, considering the negative imaginary part, is \( \arctan(\frac{-\sqrt{2}}{2}) \approx -35.264° \). Since the point is in the fourth quadrant, the principal angle is approximately 324.736°. The polar form is approximately \( 2.45 (cos 324.736° + i sin 324.736°) \).

Q3: Represent graphically and give the rectangular form of 6(cos 180° + i sin 180°)

The expression corresponds to a point on the negative real axis with magnitude 6 and angle 180°. In rectangular form, this simplifies to \( 6 (\cos 180° + i \sin 180°) = 6 (-1 + 0i) = -6 \).

Q4: Represent graphically and give the rectangular form of 7.32 − 270°

Here, the magnitude r is 7.32, and the angle θ is −270°, which is coterminal with 90°. Since \( \cos 90° = 0 \) and \( \sin 90° = 1 \), the rectangular form is \( 7.32 (0 + i \times 1) = 0 + 7.32i \).

Q5: Convert the complex number 5 + 2i to polar form

Calculate \( r = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.385 \). For the angle, \( \theta = \arctan(\frac{2}{5}) \approx 21.801° \). The polar form is approximately \( 5.385 (cos 21.801° + i sin 21.801°) \).

These conversions from rectangular to polar and vice versa facilitate various operations on complex numbers, such as multiplication, division, and finding powers and roots, which are often simpler in polar form.

References

  • Adams, R. A., & Essex, C. (2013). Calculus: A Complete Course. Pearson.
  • Brown, J. W., & Churchill, R. C. (2009). Complex Variables and Applications. McGraw-Hill Education.
  • Lay, D. C. (2012). Linear Algebra and Its Applications. Pearson.
  • Rudin, W. (1987). Real and Complex Analysis. McGraw-Hill.
  • Taylor, J. R. (2005). An Introduction to Measure and Integration. American Mathematical Society.
  • Marsden, J. E., & Tromba, A. J. (2003). Vector Calculus. W. H. Freeman.
  • Krantz, S. G. (1992). The Reality of Complex Numbers. American Mathematical Monthly.
  • Churchill, R. V., & Brown, J. W. (2009). Complex Variables and Applications. McGraw-Hill.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.
  • Larson, R., & Edwards, B. H. (2010). Calculus. Brooks Cole.

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