Find the indicated roots. Express the results in rectangular...

Find the indicated roots. Express the results in rectangular form. Show that $\frac{1+\sin \theta+1 \cos \theta}{1+\sin \theta-i \cos \theta}=\sin \theta+i \cos \theta$ Assume that $\theta \neq \frac{3}{2} \pi+2 \pi k,$ where $k$ is an integer. Hint: Work with the left-hand side; first "rationalize" the denominator by multiplying by the quantity $$ \frac{(1+\sin \theta)+i \cos \theta}{(1+\sin \theta)+i \cos \theta} $$

Find the indicated roots. Express the results in rectangular form.
Show that $\frac{1+\sin \theta+1 \cos \theta}{1+\sin \theta-i \cos \theta}=\sin \theta+i \cos \theta$
Assume that $\theta \neq \frac{3}{2} \pi+2 \pi k,$ where $k$ is an integer. Hint: Work with the left-hand side; first "rationalize" the denominator by multiplying by the quantity
$$
\frac{(1+\sin \theta)+i \cos \theta}{(1+\sin \theta)+i \cos \theta}
$$

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, usually expressed in the form a + bi. This concept is foundational when dealing with mathematics that involves both real and imaginary components, particularly when performing arithmetic operations, solving equations, or representing them in different forms, such as rectangular or polar.

Rectangular Form

Rectangular form is the standard way of representing complex numbers as a + bi, where a represents the real part and b represents the imaginary part. Expressing complex numbers in rectangular form is particularly useful for carrying out operations like addition, subtraction, and comparing components of complex expressions.

Rationalization of Complex Expressions

Rationalization of complex expressions involves eliminating the imaginary part from the denominator of a fraction by multiplying both the numerator and the denominator by the complex conjugate of the denominator. This technique simplifies complex fractions and makes further manipulation, such as expressing the result in rectangular form, more straightforward.

Complex Conjugate

The complex conjugate of a complex number a + bi is given by a - bi. Multiplying a complex number by its conjugate results in a real number, as it eliminates the imaginary parts. This property is critical when rationalizing complex denominators, as it transforms a complex denominator into a real number, facilitating easier simplification and interpretation.

Trigonometric Functions in Complex Numbers

Trigonometric functions, such as sine and cosine, are often used in complex numbers, especially when relating rectangular and polar forms. Their inclusion provides a bridge between angular measurements and the structure of complex numbers, which is particularly useful in problems where identities and manipulation of expressions involving sine and cosine are required to simplify or express a complex number in its standard form.

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