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. $$\left(\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right) \div\left(\cos \frac{1}{10} \pi+i \sin \frac{1}{10} \pi\right)$$

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.

$$\left(\cos \frac{2}{5} \pi+i \sin \frac{2}{5} \pi\right) \div\left(\cos \frac{1}{10} \pi+i \sin \frac{1}{10} \pi\right)$$

Key Concepts

Polar Representation of Complex Numbers

Complex numbers can be expressed in polar form as r(cos ? + i sin ?), where r is the magnitude (or modulus) of the number and ? is the angle (or argument) it makes with the positive real axis. This form is particularly useful for performing multiplication and division, as the operations become simpler when handling the magnitudes and angles separately.

Division of Complex Numbers in Polar Form

When dividing two complex numbers in polar form, z1 = r1(cos ?1 + i sin ?1) and z2 = r2(cos ?2 + i sin ?2), the quotient is given by z1/z2 = (r1/r2)[cos(?1 ? ?2) + i sin(?1 ? ?2)]. This property follows directly from the laws of exponents and makes division straightforward by subtracting the angles while dividing the moduli.

Angle Subtraction in Complex Division

The process of dividing complex numbers in polar form involves subtracting the angle of the denominator from the angle of the numerator. This means that if you have angles ?1 and ?2 for the numerator and denominator respectively, the resulting angle becomes ?1 ? ?2. This step is crucial to obtain the correct orientation of the resulting complex number in the complex plane.

Conversion to Rectangular Form

After performing operations in polar form, the complex number is often converted to rectangular form, which is expressed as a + bi. This involves evaluating the cosine and sine functions at the resulting angle, so that the real part is a = r cos ? and the imaginary part is b = r sin ?. In cases where the trigonometric function values are standard, this conversion is straightforward and facilitates further algebraic manipulation or interpretation.

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