Problem 40 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=\frac{1}{2} \operatorname{cis}\left(\pi+\arctan \left(\frac{5}{12}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \( z = -\frac{6}{13} - i\frac{5}{26} \).

Step 1: Understanding the problem

We need to convert the complex number given in the polar form \(z = \frac{1}{2} \operatorname{cis}(\theta)\) to its rectangular form, \(z = a + bi\), where \( \theta = \pi + \arctan\left(\frac{5}{12}\right)\).

Step 2: Express the complex number in polar form

The complex number is given in the polar form as \( z = \frac{1}{2} \operatorname{cis}\left(\pi + \arctan\left(\frac{5}{12}\right)\right) \). This means we need to find the cosine and sine of the angle \( \pi + \arctan\left(\frac{5}{12}\right) \).

Step 3: Find \( \arctan\left(\frac{5}{12}\right)\)

Given: \( \tan\theta = \frac{5}{12} \). This corresponds to a right triangle where the opposite side is 5 and the adjacent side is 12.

Step 4: Calculate cosine and sine of the angle

In the right triangle, the hypotenuse \(h\) can be calculated as \( h = \sqrt{5^2 + 12^2} = \sqrt{169} = 13 \). Therefore, \( \cos\left(\arctan\left(\frac{5}{12}\right)\right) = \frac{12}{13} \) and \( \sin\left(\arctan\left(\frac{5}{12}\right)\right) = \frac{5}{13} \).

Step 5: Use angle addition formulas

For the angle \( \pi + \arctan\left(\frac{5}{12}\right) \), use the identities: \[ \cos(\pi + \theta) = -\cos(\theta) \, \text{and} \, \sin(\pi + \theta) = -\sin(\theta) \]Therefore:\[ \cos\left(\pi + \arctan\left(\frac{5}{12}\right)\right) = -\frac{12}{13} \]\[ \sin\left(\pi + \arctan\left(\frac{5}{12}\right)\right) = -\frac{5}{13} \]

Step 6: Calculate rectangular form

The rectangular form is given by: \[ z = \frac{1}{2}\left(\cos(\pi + \theta) + i\sin(\pi + \theta)\right) \]Substituting the values, we have:\[ z = \frac{1}{2}\left(-\frac{12}{13} - i\frac{5}{13}\right) \]Simplifying gives:\[ z = -\frac{6}{13} - i\frac{5}{26} \]

Step 7: Finalize the solution

The rectangular form of the complex number is \( z = -\frac{6}{13} - i\frac{5}{26} \).

Key Concepts

Polar to Rectangular Conversion

In mathematics, complex numbers can be presented in a couple of different forms. Two common ways are the polar form and the rectangular form. The polar form expresses a complex number with a magnitude and an angle, while the rectangular form uses real and imaginary components. When converting from polar to rectangular form, we use the trigonometric functions, cosine for the real part and sine for the imaginary part. - If a complex number is written as \(z = r \, \operatorname{cis}(\theta)\), this means \(z = r[\cos(\theta) + i\sin(\theta)]\).- Here, \ "r\" represents the magnitude (or modulus) of the complex number.- \ "\theta\" is the argument (or angle) of the complex number.To convert to rectangular form:- Compute the real part as \(r \cos(\theta)\).- Compute the imaginary part as \(r \sin(\theta)i\).Combining these two, the rectangular form becomes \(z = a + bi\), where \(a\) and \(b\) are real numbers. This process helps in simplifying complex arithmetic operations.

Angle Addition Formulas

Angle addition formulas play a critical role in trigonometry, allowing us to find the trigonometric functions of a sum or difference of angles. These formulas are especially useful when involving sums of angles in complex number calculations.For an angle expressed as the sum \(\phi + \theta\), here are the desired trigonometric identities:

• \(\cos(\phi + \theta) = \cos(\phi)\cos(\theta) - \sin(\phi)\sin(\theta)\)
• \(\sin(\phi + \theta) = \sin(\phi)\cos(\theta) + \cos(\phi)\sin(\theta)\)

A common scenario involves angles like \(\pi + \theta\). In this case, specific trigonometric adjustments are legitimized:

• \(\cos(\pi + \theta) = -\cos(\theta)\)
• \(\sin(\pi + \theta) = -\sin(\theta)\)

These specific formulas aid in converting the sum of these angles into usable cosine and sine values for complex number manipulations.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the trigonometric functions to one another. They're pivotal for simplifying expressions and solving equations involving trigonometric terms.Common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Quotient Identities: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• Reciprocal Identities:
  • \( \cot \theta = \frac{1}{\tan \theta} \)
  • \( \csc \theta = \frac{1}{\sin \theta} \)
  • \( \sec \theta = \frac{1}{\cos \theta} \)

In our specific problem context, understanding these identities enables us to derive cosine and sine from the given tangent using a right triangle setup.Through applying Pythagorean theorem in a right triangle with given sides, hypotenuse computations lead to precise cosine and sine values, pivotal for converting complex numbers from polar to rectangular form effortlessly.