Problem 55
Question
Multiplying or Dividing Complex Numbers Exercises \(45-56,\) perform the operation and leave the result in trigonometric form. $$\frac{12\left(\cos 92^{\circ}+i \sin 92^{\circ}\right)}{2\left(\cos 122^{\circ}+i \sin 122^{\circ}\right)}$$
Step-by-Step Solution
Verified Answer
The result of the operation is \(3(\cos -30^{\circ} + i \sin 214^{\circ})\).
1Step 1: Convert the division operation into multiplication
We rewrite the division of two complex numbers as a multiplication by using the complex conjugate of the denominator. The complex conjugate of a complex number \(z = r(\cos \theta + i \sin \theta)\) is given by \(z^* = r(\cos \theta - i \sin \theta)\). Therefore, \(z^* / z = 1\) and we can rewrite the original problem as a multiplication: \(6\left(\cos 92^{\circ}+i \sin 92^{\circ}\right) \times \frac{1}{2}(\cos 122^{\circ} - i \sin 122^{\circ})\)
2Step 2: Apply the product-to-sum formulas
We can now simplify the multiplication by applying the product-to-sum formulas. We use the formulas \(\cos( a - b) = \cos a \cos b + \sin a \sin b\) and \(\cos( a + b) = \cos a \cos b - \sin a \sin b\) for the real and imaginary parts, respectively. This gives us a complex number in trigonometric form: \(3\left(\cos \left(92^{\circ}- 122^{\circ}\right) + i \sin \left(92^{\circ} + 122^{\circ}\right)\right)\)
3Step 3: Simplify the angles
Finally, we simplify the angles in the cosine and sine functions to get the solution in trigonometric form: \(3\left(\cos -30^{\circ} + i \sin 214^{\circ}\right)\)
Key Concepts
Trigonometric Form of Complex NumbersComplex ConjugateProduct-to-Sum Formulas
Trigonometric Form of Complex Numbers
When studying complex numbers, one of the most elegant and functional representations is the trigonometric form, also known as the polar form. A complex number can be expressed using a combination of trigonometric functions, specifically cosine and sine, which relate to the number's magnitude and direction in the complex plane.
In this form, a complex number is represented as:
\[ r(\cos \theta + i \sin \theta) \]
where \( r \) is the magnitude (or modulus) of the complex number, \( \theta \) is the argument (or angle) which the line from the origin to the point makes with the positive x-axis, and \( i \) is the imaginary unit. This representation is particularly useful when multiplying or dividing complex numbers because it allows for straightforward manipulation of magnitudes and addition of angles.
In this form, a complex number is represented as:
\[ r(\cos \theta + i \sin \theta) \]
where \( r \) is the magnitude (or modulus) of the complex number, \( \theta \) is the argument (or angle) which the line from the origin to the point makes with the positive x-axis, and \( i \) is the imaginary unit. This representation is particularly useful when multiplying or dividing complex numbers because it allows for straightforward manipulation of magnitudes and addition of angles.
Complex Conjugate
In the realm of complex numbers, the concept of a complex conjugate is a fundamental one. The complex conjugate of a complex number provides a means to effectively divide complex numbers and solve equations that would otherwise be challenging.
The complex conjugate of a number \( z = a + bi \) is denoted by \( z^* \) or \( \bar{z} \) and is defined as \( z^* = a - bi \). It has the same real component but an opposite sign for the imaginary component. When a complex number is multiplied by its conjugate, the result is always a real number:
\[ z \cdot z^* = (a + bi)(a - bi) = a^2 + b^2 \]
This becomes incredibly useful when dividing complex numbers, as seen in the given exercise. By multiplying the numerator and denominator by the complex conjugate of the denominator, we transform the division into a multiplication, simplifying the problem considerably.
The complex conjugate of a number \( z = a + bi \) is denoted by \( z^* \) or \( \bar{z} \) and is defined as \( z^* = a - bi \). It has the same real component but an opposite sign for the imaginary component. When a complex number is multiplied by its conjugate, the result is always a real number:
\[ z \cdot z^* = (a + bi)(a - bi) = a^2 + b^2 \]
This becomes incredibly useful when dividing complex numbers, as seen in the given exercise. By multiplying the numerator and denominator by the complex conjugate of the denominator, we transform the division into a multiplication, simplifying the problem considerably.
Product-to-Sum Formulas
The product-to-sum formulas are trigonometric identities that convert the product of sines and cosines into a sum or difference of sines and cosines. These formulas are valuable tools in simplifying the multiplication and division of complex numbers in trigonometric form.
The relevant product-to-sum formulas are:
The relevant product-to-sum formulas are:
- \( \cos A \cos B = \frac{1}{2}[\cos(A + B) + \cos(A - B)] \)
- \( \sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)] \)
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