Question

Find each of the following in the \(x+i y\) form and compare a computer solution. $$\arcsin (3 i / 4)$$

Short answer

The solution is \( i \times \text{ln}(2) \).

Step-by-step solution

Write the formula for the inverse sine

We need to use the formula to convert \text{arcsin} to logarithmic form. The formula for \text{arcsin}(z) where z is a complex number is: \[ \text{arcsin}(z) = -i \times \text{ln}\bigg(i z + \text{sqrt}(1 - z^2)\bigg) \]

Substitute the given value

Substitute \( z = \frac{3i}{4} \) into the formula: \[ \text{arcsin}\bigg( \frac{3i}{4} \bigg) = -i \times \text{ln}\bigg(i \times \frac{3i}{4} + \text{sqrt}\bigg(1 - \bigg( \frac{3i}{4} \bigg)^2 \bigg) \bigg) \]

Simplify inside the square root

First, find the square of \( \frac{3i}{4} \): \[ \bigg( \frac{3i}{4} \bigg)^2 = \frac{9i^2}{16} = \frac{-9}{16} \] Next, plug this back into the square root expression: \[ \text{sqrt} \bigg(1 - \frac{-9}{16} \bigg) = \text{sqrt} \bigg(1 + \frac{9}{16} \bigg) = \text{sqrt} \bigg( \frac{16}{16} + \frac{9}{16} \bigg) = \text{sqrt} \bigg( \frac{25}{16} \bigg) = \frac{5}{4} \]

Simplify the argument of the logarithm

Substitute and simplify inside the logarithm: \[ i \times \frac{3i}{4} = -\frac{3}{4} \] So the argument of the logarithm is: \[ -\frac{3}{4} + \frac{5}{4} = \frac{2}{4} = \frac{1}{2} \]

Compute the logarithm

The natural logarithm of \( \frac{1}{2} \) is: \[ \text{ln}\bigg( \frac{1}{2} \bigg) = \text{ln}(1) - \text{ln}(2) = 0 - \text{ln}(2) = - \text{ln}(2) \]

Multiply by -i

Multiply the result by -i: \[ -i \times -\text{ln}(2) = i \times \text{ln}(2) \]

Set the final result

Thus, we have: \[ \text{arcsin}\bigg( \frac{3i}{4} \bigg) = i \times \text{ln}(2) \]

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse of the standard trigonometric functions, allowing us to find the angle that a given trigonometric function represents. For example, while the sine function outputs the sine of an angle, the arcsine function (denoted as \(\text{arcsin}(z)\)) outputs the angle whose sine is \(z\). When working with complex numbers, these functions are extended to the complex plane, which can introduce new challenges and necessitates a deeper understanding of their behavior.

It's important to note that, for complex numbers, the range and output of inverse trigonometric functions might not be immediately intuitive. For example, \(\text{arcsin}(3i/4)\) involves sophisticated mathematics including the use of complex logarithms to express the result.

Complex Logarithms

The logarithm of a complex number extends the notion of a logarithm from real numbers to the complex plane. The natural logarithm of a complex number \(z\) is defined as:

\[ \text{ln}(z) = \text{ln}|z| + i \theta \text{, where} \theta \text{ is the argument of } z \text{ (the angle formed with the positive real axis)} \]

When dealing with complex logarithms, we often encounter results in the form of \(a + bi\) where \(a\) and \(b\) are real numbers. This becomes crucial when solving inverse trigonometric functions for complex inputs. In our example with \(\text{arcsin}(3i/4)\), we carefully manipulated logarithmic properties to arrive at the solution in a clear step-by-step manner.

Arc Sine for Complex Numbers

The \(\text{arcsin}(z)\) function for a complex number \(z\) can be written using a logarithmic form:

\[ \text{arcsin}(z) = -i \times \text{ln}(i z + \text{sqrt}(1 - z^2)) \]

This formula allows us to calculate the arcsine of complex numbers such as \(3i/4\). In practice, we substitute the given complex number into the equation and perform step-by-step simplifications. During simplification, be mindful of the properties of imaginary and real components. For instance, squaring an imaginary number introduces a negative real component, which significantly affects the computations.

In the solution, we've seen how \(\text{arcsin}(3i/4)\) simplifies through logarithmic properties to give \(i \times \text{ln}(2)\), illustrating the power and necessity of understanding these operations.

Mathematical Problem-Solving Steps

Solving complex mathematical problems, like finding the arcsine of a complex number, involves systematic steps. Here's a breakdown of the steps used in the exercise:

1. **Formula Application**: Start with the relevant mathematical formula. For arcsine of a complex number, use \( \text{arcsin}(z) = -i \times \text{ln}(i z + \text{sqrt}(1 - z^2)) \).
2. **Substitution**: Insert the given complex value (in this case, \(3i/4\)).
3. **Simplification**: Simplify the expression within the square root.

4. **Simplifying the Argument**: Continue to simplify the argument contained within the logarithm.
5. **Logarithmic Evaluation**: Compute the natural logarithm of the simplified argument.
6. **Final Multiplication**: Multiply by \(-i\) as per the formula.
7. **Result Verification**: Confirm the final result and potentially compare with computational solutions for accuracy.

These logical and ordered steps ensure that complicated functions and operations become manageable and understandable.