Problem 53 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{34} \operatorname{cis}\left(\tan ^{-1} \frac{3}{5}\right) $$

Step-by-Step Solution

Answer

The expression is equivalent to \(5 + 3i\).

Step 1: Identify polar coordinates

Given the expression \(\sqrt{34} \operatorname{cis}\left(\tan^{-1} \frac{3}{5}\right)\), recognize it as a polar form of a complex number: \(r\operatorname{cis}(\theta)\), where \(r = \sqrt{34}\) and \(\theta = \tan^{-1} \left(\frac{3}{5}\right)\).

Step 2: Convert to rectangular form

To express the complex number in rectangular form, use the identity \(r\operatorname{cis}(\theta) = r(\cos \theta + i\sin \theta)\). This converts our polar expression to a rectangular form: \(\sqrt{34}(\cos \theta + i\sin \theta)\).

Step 3: Find \(\cos \theta\) and \(\sin \theta\)

Since \(\tan \theta = \frac{3}{5}\), place \(\theta\) in a right triangle where opposite side \(= 3\) and adjacent side \(= 5\). The hypotenuse \(h\) is calculated using Pythagoras' theorem: \(h = \sqrt{3^2 + 5^2} = \sqrt{34}\). Thus, \(\cos \theta = \frac{5}{\sqrt{34}}\) and \(\sin \theta = \frac{3}{\sqrt{34}}\).

Step 4: Calculate rectangular form components

Substitute \(\cos \theta\) and \(\sin \theta\) back into the rectangular form: \(\sqrt{34}\left(\frac{5}{\sqrt{34}} + i \frac{3}{\sqrt{34}}\right)\). Simplify to find \(5 + 3i\).

Key Concepts

Polar Coordinates

Polar coordinates are a different way of representing complex numbers, which are usually in rectangular form. They describe a complex number using a distance from the origin, known as the "magnitude" or "radius" (denoted by \(r\)), and an angle from the positive x-axis, known as the "angle" or "argument" (denoted by \(\theta\)).

In this exercise, the complex number is given in polar form as \(\sqrt{34} \operatorname{cis}(\tan^{-1} \frac{3}{5})\). Here, \(\operatorname{cis}(\theta)\) is a shorthand for \(\cos \theta + i\sin \theta\), where \(\theta\) is the angle of the complex number. The polar coordinates for the complex number are:

• Magnitude \(r = \sqrt{34}\)
• Angle \(\theta = \tan^{-1} \big( \frac{3}{5} \big)\)

Polar coordinates can be very useful, especially in complex number multiplication and division where these operations become simpler than in rectangular form.

Rectangular Form

The rectangular form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) is the real part, and \(b\) is the imaginary part of the complex number. Converting from polar to rectangular form involves using the relation \(r(\cos \theta + i\sin \theta)\).

In the provided exercise, we start with a complex number in polar form \(\sqrt{34} \operatorname{cis}(\tan^{-1} \frac{3}{5})\). To convert this to rectangular form:

• First, calculate \(\cos \theta\) and \(\sin \theta\) using the given \(\tan \theta = \frac{3}{5}\).
• This forms a right triangle with opposite side = 3, adjacent side = 5.
• Calculate the hypotenuse: \(h = \sqrt{3^2 + 5^2} = \sqrt{34}\).
• So, \(\cos \theta = \frac{5}{\sqrt{34}}\), \(\sin \theta = \frac{3}{\sqrt{34}}\).

By substituting these into the rectangular form expression \(\sqrt{34}(\cos \theta + i\sin \theta)\), we simplify to \(5 + 3i\), which is the rectangular form of the given complex number.

Trigonometric Functions

Trigonometric functions like sine, cosine, and tangent are fundamental in converting between polar and rectangular forms of complex numbers. These functions establish relationships between the angles and sides of right triangles.

- **Sine (\sin \theta)** measures the ratio of the opposite side to the hypotenuse in a right triangle.- **Cosine (\cos \theta)** measures the ratio of the adjacent side to the hypotenuse.- **Tangent (\tan \theta)** measures the ratio of the opposite side to the adjacent side.

In the context of our exercise, understanding these trigonometric functions helps in identifying \(\cos \theta = \frac{5}{\sqrt{34}}\) and \(\sin \theta = \frac{3}{\sqrt{34}}\), crucial steps when converting the complex number from polar to rectangular form. Trig functions not only simplify angle-related computations but also provide insights into the behavior of complex numbers on the complex plane.