Carry out the indicated operations. Express your results in...

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator. $$\sqrt{2}\left(\cos \frac{1}{3} \pi+i \sin \frac{1}{3} \pi\right) \times \sqrt{2}\left(\cos \frac{4}{3} \pi+i \sin \frac{4}{3} \pi\right)$$

Carry out the indicated operations. Express your results in rectangular form for those cases in which the trigonometric functions are readily evaluated without tables or a calculator.

$$\sqrt{2}\left(\cos \frac{1}{3} \pi+i \sin \frac{1}{3} \pi\right) \times \sqrt{2}\left(\cos \frac{4}{3} \pi+i \sin \frac{4}{3} \pi\right)$$

Key Concepts

Polar Form Representation of Complex Numbers

This concept involves expressing complex numbers in the form r(cos ? + i sin ?), where r is the modulus (or magnitude) and ? is the argument (or angle). This form is especially useful when dealing with multiplication, division, and finding the powers and roots of complex numbers, as the operations often become simpler in polar form.

Multiplication of Complex Numbers in Polar Form

When complex numbers are expressed in polar form, their multiplication is performed by multiplying their moduli and adding their angles. This simplifies the process significantly compared to the rectangular form, and it leverages the properties of trigonometric functions to combine the angles directly.

Conversion Between Polar and Rectangular Forms

Conversion between the polar and rectangular forms of complex numbers is crucial. To switch from polar to rectangular form, one uses the relationships x = r cos ? and y = r sin ?, where x and y are the real and imaginary parts, respectively. This conversion is often necessary for expressing the final result in a more standard or intuitive format.

Trigonometric Evaluation in Complex Analysis

Trigonometric evaluation involves calculating cosine and sine values, particularly for angles where these functions yield exact results without the need for numerical tables or a calculator. Understanding these evaluations is important for simplifying complex numbers expressed in polar form, especially when the angles are standard ones like ?/3, ?/4, or ?/6.

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