91Ó°ÊÓ

Complex numbers are numbers that have both a real part and an imaginary part. In trigonometry, they are often represented as \(a + bi\), where \(i\) is the imaginary unit satisfying \(i^2 = -1\). The imaginary unit allows for the extension of the real number system into the complex plane. Evaluating trigonometric functions with complex numbers can be a bit tricky, so let's delve into this important concept.

Example: In the exercise, we were given the function \(T(x) = 2x^2 + 1\) and asked to find \(T \left( \frac{i \sqrt{2}}{2} \right)\). Here, \(\frac{i \sqrt{2}}{2}\) is a complex number because it has a real part of 0 and an imaginary part of \(\frac{ \sqrt{2}}{2}i\).

• When substituting \(x = \frac{i \sqrt{2}}{2}\) into \(T(x)\), we must handle the imaginary unit \(i\) according to its unique properties.
• After expanding and simplifying, consider that \(i^2 = -1\) to get the correct results.

Mastering these steps will help you tackle any similar problems involving complex numbers in trigonometric contexts.

Polynomials are expressions involving variables raised to powers and represented as \ a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \. Evaluating a polynomial means finding the value of the polynomial for a given value of the variable.

Let's break down polynomial evaluation with an example from the exercise:

• The polynomial given is \(T(x) = 2x^2 + 1\).
• To evaluate \(T \left( \frac{i \sqrt{2}}{2} \right)\), substitute the given value into the polynomial.
• This step involves basic arithmetic operations and handling any special values, such as \(i\).

By carefully substituting and simplifying, we get the result step-by-step. These fundamental skills in polynomial evaluation are crucial for solving various mathematical problems.

Solving mathematical problems step-by-step ensures that each part of the process is correct and helps to understand every aspect of the problem. Let's see how this approach is applied in the exercise:

• First, identify the function to be evaluated. Here, it is \( T(x) = 2x^2 + 1 \).
• Next, substitute the given value \( x = \frac{i \sqrt{2}}{2} \) into the function.
• Then, simplify the square \( \left( \frac{i \sqrt{2}}{2} \right)^2 \).
• Multiply by the coefficient to get the intermediate result.
• Finally, add any constant terms to get the final answer.

By following these logical and structured steps, complex problems become more manageable. Practice and familiarity with this method will enhance your problem-solving skills and confidence in handling diverse mathematical exercises.