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Convert x² + (y + 8)² = 64 to a polar equation that expresses r in terms of θ.
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Ch. 5 - Complex Numbers, Polar Coordinates and Parametric Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 11
Blitzer 3rd EditionCh. 5 - Complex Numbers, Polar Coordinates and Parametric EquationsProblema 11Capitolo 5, Problema 11
Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. 2 + 2i
Guida verificata passo dopo passo1
Identify the complex number given: \(2 + 2i\), where the real part is 2 and the imaginary part is 2.
Plot the complex number on the complex plane by marking the point at coordinates \((2, 2)\), where the x-axis represents the real part and the y-axis represents the imaginary part.
Calculate the modulus (or magnitude) \(r\) of the complex number using the formula \(r = \sqrt{(\text{real part})^2 + (\text{imaginary part})^2} = \sqrt{2^2 + 2^2}\).
Find the argument (or angle) \(\theta\) of the complex number using \(\theta = \tan^{-1}\left(\frac{\text{imaginary part}}{\text{real part}}\right) = \tan^{-1}\left(\frac{2}{2}\right)\), which can be expressed in degrees or radians.
Write the complex number in polar form as \(r(\cos \theta + i \sin \theta)\) or equivalently \(r \operatorname{cis} \theta\), using the values of \(r\) and \(\theta\) calculated in the previous steps.
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Complex Numbers and the Complex Plane
A complex number is expressed as a + bi, where a is the real part and b is the imaginary part. It can be represented as a point or vector in the complex plane, with the x-axis as the real axis and the y-axis as the imaginary axis. Plotting involves locating the point (a, b) on this plane.
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4:22
Dividing Complex Numbers
Polar Form of Complex Numbers
Polar form expresses a complex number using its magnitude (distance from origin) and argument (angle with the positive real axis). It is written as r(cos θ + i sin θ) or r∠θ, where r = √(a² + b²) and θ = arctangent(b/a). This form highlights the number's geometric properties.
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04:47Complex Numbers In Polar Form
Calculating Magnitude and Argument
The magnitude r is the length of the vector from the origin to the point (a, b), calculated by r = √(a² + b²). The argument θ is the angle formed with the positive real axis, found using θ = arctan(b/a), adjusted for the correct quadrant. These values convert rectangular form to polar form.
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04:44Finding Magnitude of a Vector
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