Problem 37 - College Trigonometry

Question

Find the rectangular form of the given complex number. Use whatever identities are necessary to find the exact values. $$ z=15 \operatorname{cis}(\arctan (-2)) $$

Step-by-Step Solution

Answer

The rectangular form of the complex number is \( z = 3\sqrt{5} - 6i\sqrt{5} \).

Step 1: Understand the given form

The complex number is given in polar form as \( z = 15 \operatorname{cis}(\theta) \), where \( \theta = \arctan(-2) \). The polar form uses the magnitude \( r = 15 \) and the angle \( \theta \).

Step 2: Express \( \operatorname{cis}(\theta) \)

The notation \( \operatorname{cis}(\theta) \) represents \( \cos(\theta) + i\sin(\theta) \). So, \( z = 15(\cos(\theta) + i\sin(\theta)) \).

Step 3: Calculate \( \theta \) using \( \arctan(-2) \)

Solve \( \theta = \arctan(-2) \). This angle is the angle whose tangent is -2. It lies in the fourth quadrant because the tangent is negative.

Step 4: Find \( \cos(\theta) \) and \( \sin(\theta) \) numerically

For \( \theta \) in the fourth quadrant where tangent is -2, use the identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = -2 \). Let \( \cos(\theta) = x \) and \( \sin(\theta) = y \). Then \( \frac{y}{x} = -2 \) means \( y = -2x \). Using \( x^2 + y^2 = 1 \), replace \( y \) to get \( x^2 + (-2x)^2 = 1 \).

Step 5: Solve for \( x \) and \( y \)

Solve \( x^2 + 4x^2 = 1 \rightarrow 5x^2 = 1 \rightarrow x^2 = \frac{1}{5} \rightarrow x = \pm \frac{1}{\sqrt{5}} \). Since \( \cos(\theta) > 0 \) in the fourth quadrant, \( x = \frac{1}{\sqrt{5}} \). Use \( y = -2x \) to find \( y = \frac{-2}{\sqrt{5}} \).

Step 6: Substitute \( \cos(\theta) \) and \( \sin(\theta) \) back

The rectangular form is \( z = 15 \left( \frac{1}{\sqrt{5}} + i \frac{-2}{\sqrt{5}} \right) = \frac{15}{\sqrt{5}} + i \left( \frac{-30}{\sqrt{5}} \right) \).

Step 7: Simplify the rectangular form

Simplify \( \frac{15}{\sqrt{5}} = 15 \times \frac{\sqrt{5}}{5} = 3\sqrt{5} \) and \( \frac{-30}{\sqrt{5}} = -30 \times \frac{\sqrt{5}}{5} = -6\sqrt{5} \). Therefore, \( z = 3\sqrt{5} - 6i\sqrt{5} \).

Key Concepts

Polar Form

When dealing with complex numbers, polar form is a way of expressing these numbers using a magnitude and an angle. In our exercise, the polar form is given as \( z = 15 \operatorname{cis}(\theta) \), where \( \theta = \arctan(-2) \).

• The magnitude, \( r \), here is \( 15 \), representing the length of the vector from the origin to the point in the complex plane.
• The angle, \( \theta \), is measured from the positive x-axis in the complex plane. It determines the direction of the vector.

Polar form is convenient for operations like multiplication and division of complex numbers. This is because magnitudes can be simply multiplied and angles added or subtracted.

Rectangular Form

The rectangular form of a complex number expresses it as a sum of real and imaginary parts: \( z = x + yi \). In the exercise, our task is to transform the given polar form into rectangular form.

• Using the expression \( \operatorname{cis}(\theta) = \cos(\theta) + i \sin(\theta) \), the number becomes \( z = 15(\cos(\theta) + i \sin(\theta)) \).
• The values \( \cos(\theta) \) and \( \sin(\theta) \) need to be determined to convert to rectangular form.

By plugging in these values, you translate the complex number into the format \( z = x + yi \), making it easier to visualize on the complex plane and perform arithmetic operations.

Trigonometric Identities

To find the sine and cosine of the angle \( \theta = \arctan(-2) \), it's necessary to apply trigonometric identities. The tangent identity is used first: \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = -2 \).

• From this, set \( \cos(\theta) = x \) and \( \sin(\theta) = y \), yielding \( y = -2x \).
• Using the Pythagorean identity \( x^2 + y^2 = 1 \), substitute \( y = -2x \) to find \( x^2 + 4x^2 = 1 \).
• Solving gives \( x = \pm \frac{1}{\sqrt{5}} \), but because \( \cos(\theta) \) is positive in the fourth quadrant, \( x = \frac{1}{\sqrt{5}} \).
Ultimately, \( \sin(\theta) \) can then be calculated as \( y = \frac{-2}{\sqrt{5}} \), providing the necessary components to express the complex number in rectangular form.

Fourth Quadrant

In trigonometry, angles are often situated in one of four quadrants of the coordinate system. The fourth quadrant is the section where the x-values (or cosine) are positive, while the y-values (or sine) are negative.

• This occurs when angles are between \( 270^{\circ} \) and \( 360^{\circ} \) (or equivalently \( \frac{3\pi}{2} \) and \( 2\pi \) radians).
• In this exercise, because \( \arctan(-2) \) results in \( \theta \) being in the fourth quadrant, we determined \( \cos(\theta) \) is positive and \( \sin(\theta) \) is negative.

Knowing which quadrant an angle lands in helps make informed guesses about the signs of sine and cosine values, crucial for accurately determining a complex number's rectangular form.