Problem 36 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=\sqrt{10} \operatorname{cis}\left(\arctan \left(\frac{1}{3}\right)\right) $$

Step-by-Step Solution

Answer

The rectangular form is \( z = 3 + i \).

Step 1: Understanding the Given Form

The complex number is given in polar form as \( z = r \operatorname{cis}(\theta) \) where \( r = \sqrt{10} \) and \( \theta = \arctan \left( \frac{1}{3} \right) \). Our goal is to convert this into rectangular form, \( z = x + yi \), where \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \).

Step 2: Calculating \( \cos(\theta) \) and \( \sin(\theta) \)

Since \( \theta = \arctan \left( \frac{1}{3} \right) \), we visualize a right triangle where the opposite side is 1 and the adjacent side is 3. Using the Pythagorean theorem, the hypotenuse is \( \sqrt{1^2 + 3^2} = \sqrt{10} \). Thus, \( \cos(\theta) = \frac{3}{\sqrt{10}} \) and \( \sin(\theta) = \frac{1}{\sqrt{10}} \).

Step 3: Apply Values to Conversion Formulas

Now that we have \( \cos(\theta) = \frac{3}{\sqrt{10}} \) and \( \sin(\theta) = \frac{1}{\sqrt{10}} \), we can find \( x \) and \( y \) using the formulas \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Thus, \( x = \sqrt{10} \times \frac{3}{\sqrt{10}} = 3 \) and \( y = \sqrt{10} \times \frac{1}{\sqrt{10}} = 1 \).

Step 4: Combine to a Rectangular Form

Now combine \( x \) and \( y \) to write the complex number in rectangular form: \( z = x + yi = 3 + 1i \).

Key Concepts

Polar to Rectangular Form

When dealing with complex numbers, it's often useful to convert between polar and rectangular forms. In polar form, a complex number is expressed as \( z = r \operatorname{cis}(\theta) \), where \( r \) is the modulus and \( \theta \) is the argument of the complex number. The term \( \operatorname{cis}(\theta) \) stands for \( \cos(\theta) + i \sin(\theta) \). This makes it easier to handle calculations involving magnitudes and angles. To convert from polar to rectangular form, we use the formulas \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Here, \( x \) and \( y \) are the real and imaginary parts, respectively, of the rectangular representation \( z = x + yi \). This conversion is particularly useful in simplifying calculations and visualizing complex numbers on the complex plane. By breaking down the polar coordinates into their rectangular components, students gain a clearer understanding of complex numbers in a plane geometric context.

Trigonometric Identities

Trigonometric identities play a crucial role in converting complex numbers from polar to rectangular form. They allow us to calculate the precise values of \( \sin(\theta) \) and \( \cos(\theta) \), which are essential for conversion. For instance, the basic identities for sine and cosine in terms of a right triangle involve the opposite, adjacent sides, and the hypotenuse.

• \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)

In the example provided, \( \theta = \arctan \left( \frac{1}{3} \right) \), which makes \( \theta \) the angle in a triangle where the opposite side is 1 and the adjacent side is 3. By using the Pythagorean theorem, we determine the hypotenuse as \( \sqrt{10} \). Thus, \( \cos(\theta) \) and \( \sin(\theta) \) can be calculated as \( \frac{3}{\sqrt{10}} \) and \( \frac{1}{\sqrt{10}} \) respectively, verifying the consistency of these trigonometric identities with geometric principles.

Right Triangle Trigonometry

Right triangle trigonometry is a fundamental concept when working with angles and sides, especially in exercises like converting polar to rectangular forms. A right triangle consists of two legs and a hypotenuse, with one of the angles being exactly 90 degrees.

• The side opposite to \( \theta \) gives the measure for \( \sin(\theta) \).
• The side adjacent to \( \theta \) gives the measure for \( \cos(\theta) \).

In the mathematical scenario presented, we visualize \( \theta = \arctan \left( \frac{1}{3} \right) \) as part of a right triangle. The opposite side of this triangle is 1, and the adjacent side is 3. Using the relation \( \sin^2(\theta) + \cos^2(\theta) = 1 \) and the calculated values for these sides, the hypotenuse is confirmed to be \( \sqrt{10} \). This understanding helps students solve for \( \cos(\theta) \) and \( \sin(\theta) \), thereby connecting the abstract concept of trigonometry to tangible geometrical interpretations.