Express \((3+2 i) e^{2 i t}+(3-2 i) e^{-2 i t}\) in terms of trigonometric functions. a) \(3 \cos 2 t-4 \sin 2 t\) c) \(6 \cos 2 t-4 \sin 2 t\) b) \(3 \cos 2 t-2 \sin 2 t\) d) \(3 \sin 2 t+2 \sin 2 t\)

Complex numbers and trigonometry are deeply interconnected, and Euler's formula serves as a bridge between them. A complex number is typically expressed as \( a+bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

In trigonometry, Euler's formula provides a powerful way to represent complex numbers in polar form as \( r(\text{cos} \theta + i \text{sin} \theta) \), where \( r \) is the magnitude of the complex number and \( \theta \) is the angle it makes with the positive real axis. This form is particularly useful for expressing the multiplication and division of complex numbers, as it simplifies the process to one of manipulating magnitudes and angles.

In the exercise, the use of Euler's formula allows us to transform the exponentials into cosines and sines, thus converting a complex exponential expression into a trigonometric one which can be easier to understand and work with.

Simplifying trigonometric expressions is a fundamental skill in mathematics, especially when dealing with complex numbers. The key to simplification is the use of trigonometric identities, which are equations involving trigonometric functions that are true for all values of the involved variables.

Some common steps involved in simplification include:

• Combining like terms, as seen in the exercise when we add the cosine and sine terms separately.
• Using Pythagorean identities to transform expressions into a common function, such as writing \( \text{sin}^2(x) \) in terms of \( \text{cos}^2(x) \).
• Applying angle sum and difference identities to expand or condense expressions.
• Utilizing even-odd identities, like those applied in Step 2 of the exercise to simplify \( \text{cos}(-x) \) and \( \text{sin}(-x) \).

Through these methods, even the most complex expressions can often be broken down into simpler, more manageable parts.

Trigonometric identities are the backbone of solving many trigonometric problems. They are essentially equations that are true for all values of the variable involved and provide a set of tools for simplifying trigonometric expressions and solving equations.

Three important categories of identities include:

• Pythagorean Identities: These relate the squares of the sine and cosine functions to 1 and are derived from the Pythagorean theorem.
• Angle Sum and Difference Identities: These allow the simplification of the sine, cosine, and tangent of sum or difference angles.
• Even-Odd Identities: These are used to find the values of trigonometric functions for negative inputs and state that cosine is an even function, and sine is an odd function.

Trigonometric identities are precisely what enabled the simplification in the given exercise. Recognizing when and how to apply these identities is crucial in streamlining complex trigonometric problems into simpler terms that can be easily solved or further manipulated.