Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{10} \operatorname{cis}\left(\tan ^{-1} 3\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( 1 + 3i \).

Step 1: Understand the Expression in Polar Form

The given expression \( \sqrt{10} \operatorname{cis}\left(\tan^{-1} 3\right) \) is in polar form, where \( r = \sqrt{10} \) is the magnitude and \( \theta = \tan^{-1} 3 \) is the argument. The expression is equivalent to \( \sqrt{10} (\cos(\theta) + i \sin(\theta)) \).

Step 2: Compute the Argument Angle

Calculate \( \theta = \tan^{-1} 3 \). Using the tangent's definition, \( \tan(\theta) = \frac{3}{1} \) implies that \( \theta \) is the angle whose tangent is 3, often computed using a calculator to get \( \theta \approx 71.565° \) or \( \theta \approx 1.249 \) radians.

Step 3: Apply Trigonometric Functions

Compute \( \cos(\theta) \) and \( \sin(\theta) \) for the angle \( \theta = \tan^{-1} 3 \). Since \( an(\theta) = \frac{3}{1} \), we use a right triangle to determine that: \( \cos(\theta) = \frac{1}{\sqrt{10}} \) and \( \sin(\theta) = \frac{3}{\sqrt{10}} \).

Step 4: Multiply by the Magnitude

Substitute \( \cos(\theta) \) and \( \sin(\theta) \) back into the expression \( \sqrt{10} (\cos(\theta) + i \sin(\theta)) \). This gives us:\[\sqrt{10} \left(\frac{1}{\sqrt{10}} + i \frac{3}{\sqrt{10}}\right) = 1 + 3i\]

Step 5: Write in the Form \(a + bi\)

Finally, the expression \( \sqrt{10} \operatorname{cis}(\tan^{-1} 3) \) simplifies to \( 1 + 3i \), which is in the form \( a + bi \), where \( a = 1 \) and \( b = 3 \).

Key Concepts

Polar Form

Polar form is a way to express complex numbers using a magnitude and an angle. Instead of representing a complex number as a sum of real and imaginary parts, polar form uses the format: \( r \times ( ext{cos}\theta + i \text{sin}\theta) \), where\( r \) is the magnitude and \( \theta \) is the angle, known as the argument.

This form highlights the geometric nature of complex numbers, which can be visualized as points in a two-dimensional plane. Here:

• \( r \) – reflects the distance from the origin to the point.
• \( \theta \) – is the angle the line makes with the positive x-axis.

Polar form simplifies multiplication and division of complex numbers, by allowing easy handling of magnitudes and angles separately.

Trigonometric Functions

Trigonometric functions like cosine and sine play an essential role in the polar form of complex numbers. They are fundamental when converting between polar and rectangular forms.

The cosine function (\( \cos \theta \)) gives the horizontal component, and the sine function (\( \sin \theta \)) provides the vertical component of the complex number on the plane. For the expression \( \text{cis} \theta \), which stands for \( \text{cos} \theta + i \text{sin} \theta \), these trigonometric functions split the magnitude into its x and y components, helping convert back to the form \( a + bi \).

Understanding these functions is key to visualizing and working with angles and sides in trigonometry, especially when dealing with right triangles or any geometric shapes connected with angles and lengths.

Right Triangle

Right triangles lie at the heart of trigonometry and the conversion processes involving polar forms. When dealing with the trigonometric interpretations of complex numbers, right triangles help us determine the necessary functions – sine, cosine, and tangent.

Given the function \( \tan \theta = \frac{3}{1} \), this represents a right triangle where \( 3 \) is the side opposite \( \theta \) and \( 1 \) is the adjacent side. Through the Pythagorean theorem, the hypotenuse, labeled as \( r \), is calculated as \( \sqrt{1^2 + 3^2} = \sqrt{10} \).

This framework enables easy computation of trigonometric values:

• \( \cos \theta = \frac{1}{\sqrt{10}} \)
• \( \sin \theta = \frac{3}{\sqrt{10}} \)

These components allow us to construct the complex number in Cartesian form from its polar counterpart.

Imaginary Unit

The imaginary unit, denoted by \( i \), is the foundational element for imaginary and complex numbers, defined as \( i = \sqrt{-1} \). In the context of complex numbers, \( i \) separates the real and imaginary components.

In expressions like \( a + bi \), \( a \) and \( b \) are real numbers, while \( i \) notes the imaginary part. When complex numbers are expressed in polar form, the imaginary unit still functions to ensure that the sine component of the angle remains orthogonal to the real part, involving the cosine.

In our solution with \( \sqrt{10} \text{cis}(\tan^{-1}3) = 1 + 3i \), \( i \) ensures that the value \( 3 \) is correctly managed as part of the imaginary portion, illustrating the interplay between real and imaginary manifestations of complex numbers.