Problem 55 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 47-56: Express in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \sqrt{5} \operatorname{cis}\left[\tan ^{-1}\left(-\frac{1}{2}\right)\right] $$

Step-by-Step Solution

Answer

The expression in rectangular form is \(2 - i\).

Step 1: Understanding the Problem

The given expression is in the polar form \( r \operatorname{cis}(\theta) \), where \( r = \sqrt{5} \) and \( \theta = \tan^{-1} \left( -\frac{1}{2} \right) \). We need to convert this into the rectangular form \( a + bi \).

Step 2: Calculating the Angle \( \theta \)

The angle \( \theta = \tan^{-1} \left( -\frac{1}{2} \right) \) is the inverse tangent of \(-\frac{1}{2}\). We know \( \tan(\theta) = -\frac{1}{2} \) and the angle corresponds to \( \theta \) in the fourth quadrant, since the tangent is negative in the fourth quadrant.

Step 3: Converting to Rectangular Form

To convert the polar expression to rectangular form, use the identities \( \cos(\theta) = \frac{1}{\sqrt{1 + \left(\frac{1}{2}\right)^2}} = \frac{2}{\sqrt{5}} \) and \( \sin(\theta) = -\frac{1/2}{\sqrt{1 + \left(\frac{1}{2}\right)^2}} = -\frac{1}{\sqrt{5}} \).

Step 4: Multiplying by the Modulus

Now multiply these trigonometric values by the modulus \( \sqrt{5} \). This gives us: - \( a = \sqrt{5} \times \frac{2}{\sqrt{5}} = 2 \)- \( b = \sqrt{5} \times \left(-\frac{1}{\sqrt{5}}\right) = -1 \).

Step 5: Writing Final Expression

Substitute the calculated values of \( a \) and \( b \) into the form \( a + bi \). Thus, the expression in rectangular form is \( 2 - i \).

Key Concepts

Polar Coordinates

Polar coordinates provide a way to denote points on a plane using a radius and an angle. In this system:

• The radius (or modulus), denoted as \( r \), represents the distance of the point from the origin.
• The angle (or argument), denoted as \( \theta \), signifies the direction from the origin to the point, measured in radians from the positive x-axis.

This method of describing points is different from the regular Cartesian coordinates that use x and y values.
Polar coordinates are particularly useful in scenarios involving circular or spiral patterns, offering a more intuitive way to describe points when angles are involved.
A complex number in polar form is often written as \( r \text{cis}(\theta) \), where \( \text{cis}(\theta) \) represents \( \cos(\theta) + i\sin(\theta) \). Here, the transformation of the polar form to a rectangular form involves trigonometric calculations of cosine and sine using the angle \( \theta \).

Rectangular Form

Rectangular form or Cartesian form of a complex number is expressed as \( a + bi \), where:

• \( a \) is the real part.
• \( b \) is the imaginary part.

This form directly represents a point on the complex plane, equivalent to a point in a traditional Cartesian coordinate system.
To convert a complex number from polar to rectangular form, you use the trigonometric identities to calculate \( a \) and \( b \):

• \( a = r \cos(\theta) \)
• \( b = r \sin(\theta) \)

By applying these identities to the polar form, you switch from using a radius and angle to using horizontal and vertical coordinates.
This conversion is crucial for simplifying complex arithmetic, especially when adding, subtracting, or comparing complex numbers.

Inverse Trigonometric Functions

Inverse trigonometric functions help us determine the angles given the value of a trigonometric ratio. For a function like the inverse tangent, \( \tan^{-1} \), the input is the ratio of the opposite side to the adjacent side in a right triangle.
Given \( \tan^{-1} \left( -\frac{1}{2} \right) \), we seek the angle which has tangent equal to \( -\frac{1}{2} \). This function returns angles typically measured in radians and is restricted to ranges where the original trigonometric function is continuous and one-to-one.
For example:

• \( \tan^{-1} \) gives angles between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \), typically in the first and fourth quadrants.

Understanding the output angle's quadrant is crucial since it affects the sign and specific values of \( \cos(\theta) \) and \( \sin(\theta) \), especially when converting to rectangular form.

Quadrants in Trigonometry

Trigonometry divides the plane into four quadrants, influencing the sign and value of trigonometric functions:

• Quadrant I is where both x and y are positive, so \( \cos \) and \( \sin \) are positive.
• Quadrant II is where x is negative and y is positive, so \( \cos \) is negative and \( \sin \) is positive.
• Quadrant III is where both x and y are negative, so both \( \cos \) and \( \sin \) are negative.
• Quadrant IV is where x is positive and y is negative, so \( \cos \) is positive and \( \sin \) is negative.

The quadrant of an angle affects the sign of the calculated sine and cosine values and determines the correct conversion from polar to rectangular form.
For \( \tan^{-1} \left( -\frac{1}{2} \right) \), the angle falls into the fourth quadrant since the tangent value is negative in that quadrant. Understanding which quadrant you're working in helps prevent mistakes in determining the correct signs for real and imaginary parts of complex numbers.