Question

Find each of the following in the $x+iy$ form and check your answers by computer. $$\cosh (i\pi +2)$$

Short answer

-$\frac{e^2+e^{-2}}{2}$

Step-by-step solution

Understand the definition of Hyperbolic Cosine

The hyperbolic cosine function $\text{cosh}(z)$ is defined as $$\text{cosh}(z)=\frac{e^z+e^{-z}}{2}$$

Substitute the given value

Given $\text{cosh}(i\pi +2)$, substitute $z=i\pi +2$ into the definition: $$\text{cosh}(i\pi +2)=\frac{e^{i\pi +2}+e^{-(i\pi +2)}}{2}$$

Simplify the exponentials

Recall that $e^{a+b}=e^a\cdot e^b$. Apply this property: $$e^{i\pi +2}=e^{i\pi }{e}^2=(-1)e^2$$ $$e^{-(i\pi +2)}=e^{-i\pi }{e}^{-2}=(-1)e^{-2}$$

Substitute the simplified exponentials

Now substitute $e^{i\pi +2}$ and $e^{-(i\pi +2)}$ back into the cosine definition: $$\text{cosh}(i\pi +2)=\frac{(-1)e^2+(-1)e^{-2}}{2}=\frac{-e^2-e^{-2}}{2}$$

Combine and simplify

Combine the terms and simplify: $$\text{cosh}(i\pi +2)=-\frac{e^2+e^{-2}}{2}$$

Key concepts

hyperbolic cosine

The hyperbolic cosine function, denoted as $\text{cosh}(z)$, is one of the six main hyperbolic functions. It is defined mathematically as $$\text{cosh}(z)=\frac{e^z+e^{-z}}{2}$$, where $e$ is Euler's number, approximately equal to 2.718. The function is similar to the trigonometric cosine function but is based on exponential functions.

In the given problem, we are tasked to find the hyperbolic cosine of $i\pi +2$. This involves substituting $z=i\pi +2$ into the definition of $\text{cosh}(z)$.
Here's how it works step by step:

• First, recall the definition of hyperbolic cosine.
• Substitute $i\pi +2$ into the formula.
• Simplify the exponential terms using the property $e^{a+b}=e^a\cdot e^b$.
• Combine and further simplify the expression to get the final result.

Understanding the underlying concepts of hyperbolic cosine will help you approach similar problems with confidence.

complex numbers

Complex numbers are fundamental in many fields of mathematics and engineering. They extend the concept of one-dimensional real numbers to a two-dimensional plane by introducing an imaginary unit $i$, where $i^2=-1$.

A complex number is generally represented as $z=x+iy$, where $x$ and $y$ are real numbers, and $i$ is the imaginary unit. In the given problem, we deal with the complex number $z=i\pi +2$, blending both real and imaginary parts.

Key properties of complex numbers include:

• The real part is $Re(z)=x$.
• The imaginary part is $Im(z)=y$.
• The magnitude (or modulus) is given by $|z|=\sqrt{x^2+y^2}$.
• The argument (or phase) is $\theta =\text{arg}(z)$.

In our problem, we use these properties to simplify the exponential terms $e^{i\pi }$ and $e^{-i\pi }$. Understanding complex numbers aids in tackling various mathematical problems and helps in deciphering the intricacies in your homework.

mathematical methods

Mathematical methods are systematic approaches used to solve problems or demonstrate theories. In dealing with the hyperbolic cosine of a complex number, these methods often involve a sequence of well-defined steps.

Here is a brief summary of the methods we used in our problem:

• Substitution: Replacing variables with actual values.
• Simplification: Using algebraic identities and properties to reduce expressions.
• Exponential properties: Applying properties like $e^{a+b}=e^a{e}^b$ to handle complex exponents.
• Combination: Combining like terms to form a more concise result.

These methods ensure our solutions are streamlined and correct. They also help maintain accuracy and consistency when solving complex mathematical problems. By understanding these methods, you can apply them not only to hyperbolic functions but to a broad range of mathematical topics, ensuring a deeper grasp of math concepts overall.