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. $$\sqrt{6}\left(\cos \frac{16}{9} \pi+i \sin \frac{16}{9} \pi\right) \div \sqrt{2}\left(\cos \frac{1}{9} \pi+i \sin \frac{1}{9} \pi\right)$$

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.

$$\sqrt{6}\left(\cos \frac{16}{9} \pi+i \sin \frac{16}{9} \pi\right) \div \sqrt{2}\left(\cos \frac{1}{9} \pi+i \sin \frac{1}{9} \pi\right)$$

Key Concepts

Complex Numbers

Complex numbers extend the real numbers by including the square root of negative one, denoted as i. They are typically expressed in rectangular form as a + bi, where a and b are real numbers, or in polar form as a magnitude (or modulus) and an angle (or argument). This dual representation allows for operations such as addition, multiplication, division, and exponentiation to be performed in the most efficient manner depending on the context.

Polar Form Representation

The polar form of a complex number expresses it as r(cos ? + i sin ?), where r is the modulus and ? is the argument. This form is particularly useful for multiplication and division because the modulus and arguments can be manipulated separately. It provides a geometric interpretation of complex numbers as points or vectors in the plane, making it easier to understand their behavior under various operations.

Division of Complex Numbers in Polar Form

When dividing complex numbers expressed in polar form, the moduli (magnitudes) are divided and the arguments (angles) are subtracted. This rule simplifies the division process by reducing it to straightforward arithmetic operations on the magnitudes and angles, thus avoiding the need to perform more complicated algebraic manipulations in the rectangular form.

Conversion Between Polar and Rectangular Forms

The conversion between polar and rectangular forms is essential for expressing complex numbers in the desired format. To convert a number from polar to rectangular form, one uses the relationships a = r cos ? and b = r sin ?. This is particularly useful when the result of a computation in polar form needs to be interpreted or further manipulated in standard algebraic operations typically carried out in rectangular coordinates.

Trigonometric Evaluations

Trigonometric evaluations involve determining the cosine and sine of specific angles, which are often encountered in polar form representations of complex numbers. In cases where these angles are multiples of common fractions of ?, the resulting sine and cosine values can be computed exactly, making it possible to express the final result in an exact rectangular form without needing numerical approximations.

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