Problem 111 - Precalculus with Limits

Question

Think About It Show that the negative of \(z=r(\cos \theta+i \sin \theta)\) is $$-z=r[\cos (\theta+\pi)+i \sin (\theta+\pi)]$$

Step-by-Step Solution

Answer

The negative of a complex number \(z= r(\cos \theta + i \sin \theta)\) can indeed be expressed as \(-z= r[\cos (\theta+\pi)+i \sin(\theta +\pi)]\) as proven in the steps above.

Step 1: Identify the negative of \(z\)

To find the negative of a given complex number \(z\), simply put a negative sign in front of it. This yields \(-z= -r(\cos \theta+i \sin \theta)\).

Step 2: Use trigonometric identities

Recall the trigonometric identities \(\cos(\theta + \pi) = - \cos( \theta)\) and \(\sin(\theta + \pi)= -\sin(\theta)\). Substituting these identities into \(-z= -r(\cos \theta+i \sin \theta)\), you get \(-z= r(\cos(\theta + \pi) +i \sin(\theta +\pi)).\)

Step 3: Verify the expression

Compare the derived expression with the one given in the problem statement. You will notice that they match perfectly, hence the original statement is verified.

Key Concepts

Trigonometric identities

Understanding trigonometric identities is crucial when working with complex numbers in trigonometric form. These identities are mathematical equations that relate the trigonometric functions sine, cosine, and tangent to one another.

Two of the most fundamental identities used for complex numbers in trigonometric form are the angle addition identities for cosine and sine:

• \(\boldsymbol{\text{cos}}(\theta + \phi) = \boldsymbol{\text{cos}}(\theta)\boldsymbol{\text{cos}}(\phi) - \boldsymbol{\text{sin}}(\theta)\boldsymbol{\text{sin}}(\phi)\)
• \(\boldsymbol{\text{sin}}(\theta + \phi) = \boldsymbol{\text{sin}}(\theta)\boldsymbol{\text{cos}}(\phi) + \boldsymbol{\text{cos}}(\theta)\boldsymbol{\text{sin}}(\phi)\)

In the context of the exercise we're discussing, we applied the angle addition identities specifically for the cases where \(\phi = \pi\), which gives:

• \(\boldsymbol{\text{cos}}(\theta + \pi) = -\boldsymbol{\text{cos}}(\theta)\)
• \(\boldsymbol{\text{sin}}(\theta + \pi) = -\boldsymbol{\text{sin}}(\theta)\)

These identities are paramount because they allow us to rewrite the negative of a complex number in a form that reveals the underlying geometry; the negation corresponds to a rotation of \(\pi\) radians (or 180 degrees) around the origin in the complex plane.

Negative of a complex number

In the realm of complex numbers, the negative of a number involves reversing the direction of the vector representation on the complex plane. For any complex number \(z = r(\cos \theta + i \sin \theta)\), its negative \(-z\) simply changes the sign of both the real and imaginary parts. Visually, this represents a 180-degree turn around the origin.

To express this turn mathematically, we utilize the previously mentioned trigonometric identities. This transformation ensures that the magnitude remains the same, but the angle becomes \(\theta + \pi\), highlighting the direct relationship between complex number operations and rotations in the complex plane.

Precalculus concepts

Precalculus serves as the foundation that bridges algebra with the more advanced topics of calculus. It involves studying various mathematical concepts including functions, complex numbers, and the trigonometric identities discussed earlier.

Concepts such as exponential, logarithmic, and trigonometric functions are key for understanding the behavior of complex numbers. As seen in our exercise, knowing how to convert complex numbers into trigonometric form and vice versa opens up a new way of visualizing mathematical operations as geometric transformations.

Additionally, precalculus covers the crucial idea of the unit circle, which helps in understanding angles and their corresponding sine and cosine values—all of which are essential in the study of complex numbers and their geometric interpretations in the complex plane.