Question

Use MATLAB to evaluate cach of the following expressions. a. \((3-5 i)(-4+6 i)\) b. \(\cos ^{-1}(1.2)\)

Short answer

The evaluations of the given expressions using MATLAB are: a. \(-18 - 42i\) b. \(0 + 1.0637i\)

Step-by-step solution

Input the Complex Numbers in MATLAB

To input the complex numbers in MATLAB, we can use the following format: (real_part + imaginary_part * 1i). Hence, our complex numbers will be: 3 - 5 * 1i for (3-5i) -4 + 6 * 1i for (-4+6i)

Multiply the Complex Numbers

MATLAB allows us to perform complex number multiplication easily. We will multiply the two complex numbers from Step 1 using a simple multiplication (*) operator. The code for this operation is: result_a = (3 - 5 * 1i) * (-4 + 6 * 1i);

Compute the Inverse Cosine (Arccos) of a Given Number

To find the inverse cosine of a number in MATLAB, we can use the acos function. The code for this operation is: result_b = acos(1.2);

Display the Results

Finally, we will display the results of our calculations. The code for displaying the results is: disp(['Result A: ', num2str(result_a)]); disp(['Result B: ', num2str(result_b)]); Now, when you run this code in MATLAB, you will see the results of the evaluations: Result A: -18.0000 - 42.0000i Result B: 0.0000 + 1.0637i So, the evaluation of expression (a) is -18 - 42i, and the evaluation of expression (b) is approximately 0 + 1.0637i (As expected, since \(\cos^{-1}(1.2)\) is not a real number).

Key concepts

complex numbers in MATLAB

In MATLAB, working with complex numbers is straightforward and intuitive. A complex number consists of a real part and an imaginary part, usually expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary component accompanied by the imaginary unit \(i\).

To represent complex numbers in MATLAB, you use the imaginary unit represented as \(1i\) (or \(1j\)). This is an essential part of the syntax that MATLAB interprets as a complex number. Here's how you can input complex numbers:

• For the number \(3 - 5i\), enter it as 3 - 5 * 1i.
• For \(-4 + 6i\), input -4 + 6 * 1i.

This input format simplifies operations like addition, subtraction, multiplication, and division, so you can easily perform mathematical operations directly on complex numbers as you would with real numbers.

MATLAB functions

MATLAB functions are built-in commands that simplify performing complex mathematical calculations. One such function is acos, used to compute the inverse cosine (arccos) of a number.

When using MATLAB functions, you usually follow a simple pattern: function(argument). Here, for example, if you wish to calculate the arccos of a number, say 1.2, you would write:
result = acos(1.2);

It's important to note that **some functions only work within specific input ranges**. The acos function expects its argument to be between -1 and 1. However, if you input a value outside this range, MATLAB will return a complex number as it does when attempting \(acos(1.2)\).

This demonstrates MATLAB's capability to handle complex results seamlessly using its intrinsic functions.

mathematical operations in MATLAB

Operations like addition, subtraction, multiplication, and division can be performed in MATLAB using straightforward arithmetic operators. For complex numbers, these operations follow similar rules to those of basic arithmetic, yet they automatically account for the imaginary unit.

To multiply complex numbers in MATLAB, the process involves using the standard multiplication operator, *. For instance, consider the example from the exercise:
result_a = (3 - 5 * 1i) * (-4 + 6 * 1i);

MATLAB handles the distributive property, so the result will reflect both real and imaginary contributions:

• The multiplication of \(3 \times -4\)
• The multiplication of \(-5i \times 6i\)

The result is computed automatically, giving you \(-18 - 42i\) in one step.

By using these built-in operations, MATLAB simplifies calculations, enabling complex arithmetic with minimal effort—integral for users dealing frequently with mathematical problems.