Question

\(2.8 e^{-i(1.1)}\)

Short answer

1.2701 - 2.4954i

Step-by-step solution

Understand the Expression

Identify that we need to simplify the complex number expression, which is given in the form of Euler's formula, where the number is multiplied by an exponential with an imaginary exponent.

Recall Euler's Formula

Euler's formula states that for any real number \theta, \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\). In this case, \theta = 1.1.

Apply Euler's Formula

Substitute \theta = 1.1 into Euler's formula: \(e^{-i(1.1)} = \cos(-1.1) + i\sin(-1.1)\). Note that \cos\ is an even function and \sin\ is an odd function, so \(\cos(-1.1) = \cos(1.1)\) and \(\sin(-1.1) = -\sin(1.1)\).

Evaluate Trigonometric Functions

Calculate \(\cos(1.1)\) and \(\sin(1.1)\) using a calculator or trigonometric table to find: \(\cos(1.1)\ \approx 0.453596\) and \(\sin(1.1)\ \approx 0.891207\).

Substitute Values

Substitute the values back into the expression: \(e^{-i(1.1)} = 0.453596 - i(0.891207)\).

Multiply by 2.8

Multiply the real and imaginary parts by 2.8: \(2.8 \times 0.453596 = 1.2700688\) and \(2.8 \times -0.891207 = -2.4953796\).

Combine Results

Combine the results to get the final simplified expression: \(2.8 e^{-i(1.1)} = 1.2701 - 2.4954i\).

Key concepts

Euler's formula

Euler's formula is a fundamental bridge between complex numbers and trigonometry. It states that for any real number \(\theta\), the exponential function with an imaginary exponent can be represented as: \[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \] Here, \(i\) is the imaginary unit, where \(i^2 = -1\). This formula shows a profound relationship by expressing complex exponentials in terms of familiar trigonometric functions.

This is especially useful when simplifying expressions of complex numbers in exponential form. It allows us to convert these exponential expressions into a sum of real and imaginary parts, making calculations more intuitive and manageable.
In our exercise, we used Euler's formula to transform the given expression \(e^{-i(1.1)}\) into trigonometric terms. By substituting 1.1 for \(\theta\) and acknowledging the even and odd properties of cosine and sine, we derived an easier format to work with.

Trigonometric Functions

Trigonometric functions, primarily cosine and sine, play a crucial role in Euler's formula. Let's dive deeper into the properties we used:

1. **Cosine (\(\cos\)) Function**
- Cosine is an even function. This means that \(\cos(-\theta) = \cos(\theta)\).

2. **Sine (\(\sin\)) Function**
- Sine is an odd function. This means that \(\sin(-\theta) = -\sin(\theta)\).

When we apply these properties to the expression \(e^{-i(1.1)}\), it simplifies our calculations:
- \(\cos(-1.1)\) remains \(\cos(1.1)\).
- \(\sin(-1.1)\) becomes \(-\sin(1.1)\).

Knowing these properties allows us to confidently manipulate and simplify expressions involving trigonometric functions.

Imaginary Exponent

An imaginary exponent, such as \(e^{i\theta}\), can initially seem puzzling. However, thanks to Euler's formula, we can express it in a more understandable form using trigonometry. The key is recognizing that: \[ e^{i\theta} = \cos(\theta) + i\sin(\theta) \]

In our case, the expression \(e^{-i(1.1)}\) involves an imaginary exponent with a negative angle.
By substituting it into Euler's formula, we get:
\[ e^{-i(1.1)} = \cos(-1.1) + i\sin(-1.1) \]

Using trigonometric identities:
- \(\cos(-1.1) = \cos(1.1)\)
- \(\sin(-1.1) = -\sin(1.1)\)

This transformation is powerful as it enables the combination of complex exponential functions into simpler, familiar trigonometric components. Ultimately, this step bridges abstract concepts with practical, calculable values, making the operations with complex numbers clearer.