📈college algebra review

key term - R(cos θ + i sin θ)

Definition

The expression r(cos θ + i sin θ) is a fundamental representation of complex numbers in polar form. It allows for the concise expression of a complex number using a magnitude (r) and an angle (θ), providing an alternative to the more familiar Cartesian form (a + bi).

5 Must Know Facts For Your Next Test

  1. The magnitude (r) in the polar form representation of a complex number is the distance from the origin to the point on the complex plane that represents the complex number.
  2. The angle (θ) in the polar form representation is the angle, measured in radians, between the positive real axis and the line segment connecting the origin to the point on the complex plane that represents the complex number.
  3. The polar form of a complex number provides a more intuitive understanding of the number's magnitude and direction compared to the Cartesian form.
  4. Conversions between the Cartesian form (a + bi) and the polar form (r(cos θ + i sin θ)) can be performed using the formulas: r = √(a^2 + b^2) and θ = tan^-1(b/a).
  5. The polar form representation is particularly useful for performing operations such as multiplication, division, and exponentiation of complex numbers, as these operations become simpler in the polar form.

Review Questions

  • Explain the relationship between the magnitude (r) and the angle (θ) in the polar form representation of a complex number.
    • The magnitude (r) in the polar form representation of a complex number, r(cos θ + i sin θ), represents the distance from the origin to the point on the complex plane that represents the complex number. The angle (θ), measured in radians, represents the angle between the positive real axis and the line segment connecting the origin to the point on the complex plane. Together, the magnitude and angle provide a more intuitive understanding of the complex number's magnitude and direction compared to the Cartesian form (a + bi).
  • Describe the process of converting a complex number from Cartesian form to polar form.
    • To convert a complex number from Cartesian form (a + bi) to polar form (r(cos θ + i sin θ)), you can use the following formulas: r = √(a^2 + b^2) θ = tan^-1(b/a) The magnitude (r) is calculated as the square root of the sum of the squares of the real and imaginary parts (a and b, respectively). The angle (θ) is calculated as the inverse tangent of the ratio of the imaginary part (b) to the real part (a). These conversions allow you to represent the complex number using its magnitude and angle, which can be more intuitive and useful for certain operations.
  • Explain how the polar form representation of complex numbers can be advantageous for performing operations such as multiplication, division, and exponentiation.
    • The polar form representation of complex numbers, r(cos θ + i sin θ), is particularly advantageous for performing operations such as multiplication, division, and exponentiation. In the polar form: - Multiplication of two complex numbers involves adding their angles and multiplying their magnitudes: (r1(cos θ1 + i sin θ1)) × (r2(cos θ2 + i sin θ2)) = (r1r2)(cos (θ1 + θ2) + i sin (θ1 + θ2)). - Division of two complex numbers involves subtracting their angles and dividing their magnitudes: (r1(cos θ1 + i sin θ1)) / (r2(cos θ2 + i sin θ2)) = (r1/r2)(cos (θ1 - θ2) + i sin (θ1 - θ2)). - Exponentiation of a complex number involves raising the magnitude to the power and multiplying the angle by the power: (r(cos θ + i sin θ))^n = r^n(cos (nθ) + i sin (nθ)). These simplified operations in the polar form make it a powerful representation for performing complex number calculations.

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