Problem 109
Question
Reasoning Show that \(\overline{z}=r[\cos (-\theta)+i \sin (-\theta)]\) is the complex conjugate of \(z=r(\cos \theta+i \sin \theta)\)
Step-by-Step Solution
Verified Answer
Hence, we show that \( r[\cos (-\theta)+i\sin (-\theta)]\) is the complex conjugate of \(r(\cos \theta +i\sin \theta)\) which confirms that the given equation is indeed the complex conjugate in its polar form.
1Step 1: Definition
A complex number \(z\) in polar form is represented as \(z = r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude (or modulus) of \(z\), \(\theta\) is the angle formed with the real axis in the complex plane, \(i\) is the imaginary unit (i.e., \(i^2 = -1\)), and \(\cos\) & \(\sin\) are the cosine & sine functions.
2Step 2: Complex Conjugate
The complex conjugate of \(z\), denoted as \(\overline{z}\), is found by changing the sign of its imaginary part. This means keeping the real part intact while negating the imaginary part. Here, \(\cos \theta\) is the real part and \(i \sin \theta\) is the imaginary part. Therefore, the complex conjugate, \(\overline{z}= r[\cos \theta - i \sin \theta]\).
3Step 3: Trigonometric identity
However, \(-\sin \theta = \sin (-\theta)\). Adding this trigonometric identity to the above equation, we get \(\overline{z}= r[\cos \theta + i \sin (-\theta)]\), which matches with the given equation, proving that it is indeed the complex conjugate of \(z\).
Key Concepts
Polar FormComplex ConjugateTrigonometric Identities
Polar Form
In mathematics, particularly in complex numbers, the polar form is a way of representing a complex number using its magnitude and angle. A complex number in this form is written as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument) from the positive real axis in the complex plane.
Understanding the polar form is significant because it simplifies the multiplication, division, and powers of complex numbers. The magnitude \( r \) represents the distance from the origin to the point in the complex plane. The angle \( \theta \) represents the counterclockwise angle from the positive real axis to the line segment representing the complex number.
One reason polar forms are valuable is their ability to calculate powers and roots easily using De Moivre’s Theorem. For instance, multiplying two complex numbers in polar form is straightforward as it involves multiplying their magnitudes and adding their angles. This makes calculations more intuitive and often simpler than performing multiplications in rectangular form.
Understanding the polar form is significant because it simplifies the multiplication, division, and powers of complex numbers. The magnitude \( r \) represents the distance from the origin to the point in the complex plane. The angle \( \theta \) represents the counterclockwise angle from the positive real axis to the line segment representing the complex number.
One reason polar forms are valuable is their ability to calculate powers and roots easily using De Moivre’s Theorem. For instance, multiplying two complex numbers in polar form is straightforward as it involves multiplying their magnitudes and adding their angles. This makes calculations more intuitive and often simpler than performing multiplications in rectangular form.
Complex Conjugate
A complex conjugate is a crucial concept in complex number theory. It involves switching the sign of the imaginary part of a complex number. So, if you start with a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \).
In terms of polar coordinates, the complex conjugate of \( z = r(\cos \theta + i \sin \theta) \) can be expressed as \( \overline{z} = r(\cos \theta - i \sin \theta) \). You simply negate the imaginary component. This operation is geometrically recognized as reflecting the complex number across the real axis in the complex plane.
The complex conjugate is vital for various applications such as simplifying division and finding the modulus of complex numbers. When you multiply a complex number by its conjugate, the result is a real number represented as the square of the modulus, \( r^2 \). This property is used in the expression of functional terms in equations and is fundamental in fields involving complex number calculations.
In terms of polar coordinates, the complex conjugate of \( z = r(\cos \theta + i \sin \theta) \) can be expressed as \( \overline{z} = r(\cos \theta - i \sin \theta) \). You simply negate the imaginary component. This operation is geometrically recognized as reflecting the complex number across the real axis in the complex plane.
The complex conjugate is vital for various applications such as simplifying division and finding the modulus of complex numbers. When you multiply a complex number by its conjugate, the result is a real number represented as the square of the modulus, \( r^2 \). This property is used in the expression of functional terms in equations and is fundamental in fields involving complex number calculations.
Trigonometric Identities
Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable. In the context of complex numbers, they are valuable in simplifying expressions and solving equations.
One pivotal identity used in complex number calculations is that of the sine function, specifically \( \sin(-\theta) = -\sin(\theta) \). Applying this identity helps transform expressions in complex conjugate solutions, such as showing that \( \overline{z} = r(\cos(\theta) + i \sin(-\theta)) \).
These identities serve as the foundation for many mathematical proofs and applications, offering a bridge between algebraic and geometric interpretations of numbers. Other basic trigonometric identities include \( \cos^2(\theta) + \sin^2(\theta) = 1 \) and \( \cos(-\theta) = \cos(\theta) \). Each identity plays a crucial role in unraveling complex mathematical relationships and calculations in various fields, including engineering, physics, and computer science.
One pivotal identity used in complex number calculations is that of the sine function, specifically \( \sin(-\theta) = -\sin(\theta) \). Applying this identity helps transform expressions in complex conjugate solutions, such as showing that \( \overline{z} = r(\cos(\theta) + i \sin(-\theta)) \).
These identities serve as the foundation for many mathematical proofs and applications, offering a bridge between algebraic and geometric interpretations of numbers. Other basic trigonometric identities include \( \cos^2(\theta) + \sin^2(\theta) = 1 \) and \( \cos(-\theta) = \cos(\theta) \). Each identity plays a crucial role in unraveling complex mathematical relationships and calculations in various fields, including engineering, physics, and computer science.
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