Problem 75 - Precalculus with Limits

Question

Finding a Power of a Complex Number In Exercises \(65-80\) , use DeMoivre's Theorem to find the indicated power of the complex number. Write the result in standard form. $$[5(\cos 3.2+i \sin 3.2)]^{4}$$

Step-by-Step Solution

Answer

The fourth power of [5(cos(3.2) + i sin(3.2))] in standard form is 625 cos(12.8) + 625 i sin(12.8). The actual real and imaginary components depend on the values of cos(12.8) and sin(12.8).

Step 1 Use the DeMoivre's Theorem

According to DeMoivre's theorem, \((r(\cos x + i \sin x))^n = r^n(\cos(nx) + i \sin(nx))\). From the given number, we have \(r=5\) and \(x=3.2\), so our target is \(n=4\). Applying these values to DeMoivre's theorem, we get: \((5(\cos(3.2)+i \sin(3.2)))^{4} = 5^{4} (\cos(4*3.2) + i\sin(4*3.2))\).

Step 2 Evaluate each part separately

Break down the formula and evaluate each part separately: \(5^{4} = 625\), \(\cos(4*3.2) = \cos(12.8) \), and \(i\sin(4*3.2) = i\sin(12.8)\)

Step 3 Apply values to the formula

Substitute the calculated values back into the formula: \(625 (\cos(12.8) + i\sin(12.8))\)

Step 4 Convert to the standard form

To convert it to standard form, calculate the real and imaginary parts separately as \(625 \cos(12.8)\) and \(625 i \sin(12.8)\). Hence, we have \(625 \cos(12.8) + 625 i \sin(12.8)\) as the fourth power of the given complex number in standard form.

Key Concepts

Complex Numbers

Complex numbers form the foundation of advanced mathematical concepts. They consist of a real part and an imaginary part. The standard notation for a complex number is represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
The imaginary unit \(i\) is defined as the square root of \(-1\), giving complex numbers their unique properties.

• Real Part: portion along the real axis
• Imaginary Part: portion along the imaginary axis
• Complex Conjugate: flipping the sign of the imaginary part

Together, complex numbers allow for the expression and solution of equations that do not have real solutions, such as finding the roots of negative numbers.

Standard Form

The standard form of a complex number, often denoted as \(a + bi\), is a straightforward way to express these numbers using both real and imaginary components. Writing a complex number in this form helps in performing various algebraic operations.
When using a representation like \(a + bi\), it is easier to add, subtract, and multiply complex numbers. Let's examine its features:

• Clearly distinguishes real and imaginary parts
• Makes it straightforward to perform computations
• Simplifies conversion to other forms

For instance, after using DeMoivre's Theorem, you often convert the polar representation back into standard form to maintain consistency across calculations.

Trigonometric Functions

Trigonometric functions like sine and cosine are fundamental when dealing with complex numbers, particularly in polar form. These functions help represent a complex number's position on the two-dimensional plane.

• Cosine (\(\cos\)) measures the horizontal distance from the origin
• Sine (\(\sin\)) measures the vertical distance from the origin
• Angled relations between these projections define magnitude and direction

When using DeMoivre's Theorem, multiplying these angles and their trigonometric functions becomes invaluable for computations. For example, if a complex number is expressed using cosine and sine, DeMoivre's Theorem simplifies raising it to a power by multiplying the angle.

Polar Coordinates

Polar coordinates offer an alternative way to express complex numbers, using a magnitude and an angle rather than a point on the plane.

• Magnitude (or modulus): distance from the origin
• Angle (or argument): direction measured counterclockwise from the positive real axis

These coordinates are often more intuitive for expressing rotations and scaling, which is where DeMoivre's Theorem comes into play. When expressing a complex number in polar form, you write it as \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude and \(\theta\) is the angle. This form is especially useful in calculating powers and roots of complex numbers, as it simplifies the process significantly by transforming multiplications into additions, which are easier to handle in trigonometric expressions.