Question

Find each of the following in the \(x+i y\) form and check your answers by computer. $$\sin \frac{i \pi}{2}$$

Short answer

\[ \text{sin}\left( \frac{i \pi}{2} \right) = i \left( \frac{e^{\frac{\pi}{2}} - e^{-\frac{\pi}{2}}}{2} \right). \]

Step-by-step solution

Write the given expression

The given expression to find is \(\frac{i \, \pi}{2}\). We need to find \(\text{sin} \, \frac{i \, \pi}{2}\) in the form of \(x + i y\).

Recall the definition of \(\text{sin}(z)\) for a complex number

The sine of a complex number \(z = x + i y\) where \(x\) and \(y\) are real numbers is given by:\[ \sin(z) = \sin(x + iy) = \sin(x) \cosh(y) + i \cos(x) \sinh(y) \]

Substitute \(z = \frac{i \, \pi}{2}\) into the definition

Here, \(x = 0\) and \(y = \frac{\pi}{2}\). Substituting these values in, we obtain:\[ \sin \left(0 + i \frac{\pi}{2} \right) = \sin(0) \cosh \left( \frac{\pi}{2} \right) + i \cos(0) \sinh \left( \frac{\pi}{2} \right) \]

Evaluate each trigonometric and hyperbolic function

We know from trigonometric and hyperbolic identities that:\(\sin(0) = 0\),\(\cos(0) = 1\),\(\cosh \left( \frac{\pi}{2} \right) = \frac{e^{\frac{\pi}{2}} + e^{-\frac{\pi}{2}}}{2}\),\(\sinh \left( \frac{\pi}{2} \right) = \frac{e^{\frac{\pi}{2}} - e^{-\frac{\pi}{2}}}{2}\).

Simplify the expression

By substituting these values back into the equation, we have:\[ \sin \left( \frac{i \pi}{2} \right) = 0 \cdot \cosh \left( \frac{\pi}{2} \right) + i \cdot 1 \cdot \sinh \left( \frac{\pi}{2} \right) = i \cdot \frac{e^{\frac{\pi}{2}} - e^{-\frac{\pi}{2}}}{2}\].

Final result

Recognizing that \(\sinh \left( \frac{\pi}{2} \right)\) is the same as expressing:\[ \sin \left( \frac{i \pi}{2} \right) = i \sinh \left( \frac{\pi}{2} \right) \implies \sin \left( \frac{i \pi}{2} \right) = i \left( \frac{e^{\frac{\pi}{2}} - e^{-\frac{\pi}{2}}}{2} \right). \]

Key concepts

Complex Numbers

Complex numbers are numbers that include both a real part and an imaginary part. They can be written in the form \(a + bi\) where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property \(i^2 = -1\). Complex numbers are crucial in solving problems that cannot be addressed using just real numbers. They appear in various fields including engineering, physics, and applied mathematics.
When dealing with complex functions like sine or cosine, you need to handle the real and imaginary parts separately. This allows you to unravel more complex expressions and understand behaviors in different domains.

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but for the hyperbola rather than the circle. They include hyperbolic sine (sinh) and hyperbolic cosine (cosh).
Each hyperbolic function has a specific identity, similar to trigonometric ones, and they help solve a wide array of problems involving complex numbers. For instance, the hyperbolic sine and cosine of a value \(y\) can be defined as:
\[\text{sinh}(y) = \frac{e^y - e^{-y}}{2} \]
\[\text{cosh}(y) = \frac{e^y + e^{-y}}{2} \]
These relationships come in handy when transforming and simplifying expressions involving complex arguments, as seen in our exercise with the value \(\frac{i \, \pi}{2}\). By using these identities, you can change complex trigonometric expressions into more manageable forms.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for different values of their variables. These identities are important as they enable the simplification and transformation of complex trigonometric expressions.
A few key identities used in the context of complex numbers include:

• \[\text{sin}(0) = 0 \]
  
• \[\text{cos}(0) = 1 \]
  
• \[\text{sin}(x) = \text{cosh}(y) + i \text{cos}(x) \text{sinh}(y) \]
  

By leveraging these identities, you can break down and understand more complicated mathematical concepts. These identities also serve as basic building blocks for solving problems involving complex functions like the one in our given exercise.
They connect the behavior of trigonometric functions with those of hyperbolic functions, providing a unified framework for analysis and computation.