Problem 47 - Algebra and Trigonometry Real Mathematics, Real People

Question

Represent the complex number graphically, and find the standard form of the number. $$6\left(\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}\right)$$

Step-by-Step Solution

Answer

The Cartesian form of the complex number \( 6(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \) is \( 3 + 3\sqrt{3}i \). The point representing this number on the complex plane is (3, 3\sqrt{3}).

Step 1: Convert to Cartesian Form

To convert the complex number to Cartesian form, the formula is: \( r(\cos w + i \sin w) = r \cos w + i r \sin w \). Under this rule, \( 6(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \) becomes \( 6 \cos \frac{\pi}{3} + i 6 \sin \frac{\pi}{3} \).

Step 2: Calculate Real and Imaginary Part

The real part of the solution is obtained by multiplying the magnitude of the complex number (r) with the cosine of the angle (\( \cos \) w), while the imaginary part of the solution is obtained by multiplying the magnitude of the complex number with the sine of the angle (\( \sin \) w). Therefore, the real part is \( 6 \cos \frac{\pi}{3} = 3 \) and the imaginary part is \( 6 \sin \frac{\pi}{3} = 3\sqrt{3} \). So the Cartesian Form is \( 3 + 3\sqrt{3}i \) .

Step 3: Plot the Complex Number on a Complex Plane

Plot the real part (3) on the x-axis (Real) and the imaginary part (3\sqrt{3}) on the y-axis (Imaginary). The complex number is represented by the point (3, 3\sqrt{3}) on the complex plane.

Key Concepts

Cartesian Form of Complex Numbers

The Cartesian form is a standard way to express complex numbers, consisting of a real part and an imaginary part. When we refer to complex numbers in the Cartesian form, we write them as a combination of these two components: \( a + bi \), where \( a \) is the real part, \( b \) is the imaginary part, and \( i \) is the imaginary unit, satisfying \( i^2 = -1 \).

For example, given the complex number \( 6(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \), we can convert it into the Cartesian form by identifying \( a \) as \( 6 \cos \frac{\pi}{3} \) and \( b \) as \( 6 \sin \frac{\pi}{3} \). After the calculation, we find that \( a = 3 \) and \( b = 3\sqrt{3} \), resulting in the Cartesian form of \( 3 + 3\sqrt{3}i \).

Polar to Cartesian Conversion

Polar to Cartesian conversion is used to transform a complex number from its polar form \( r(\cos \theta + i \sin \theta) \) to its Cartesian form \( a + bi \). The polar form captures the magnitude \( r \) and angle \( \theta \), also known as argument, of the complex number.

To convert from polar to Cartesian, we use the equations \( a = r \cos \theta \) for the real part, and \( b = r \sin \theta \) for the imaginary part. In our exercise, the number \( 6(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3}) \) is converted by calculating the real and imaginary parts using the given \( r \) and \( \theta \). This results in a Cartesian representation of the complex number.

Plotting Complex Numbers

Plotting complex numbers on the complex plane is like plotting coordinates on a two-dimensional graph. The horizontal axis (x-axis) represents the real part of the complex number, while the vertical axis (y-axis) represents the imaginary part.

In our exercise, the complex number \( 3 + 3\sqrt{3}i \) is plotted by locating the real part, which is 3, on the x-axis, and the imaginary part, which is \( 3\sqrt{3} \), on the y-axis. By drawing a point where these two values intersect, we can graphically represent the complex number. This visual interpretation is vital for understanding complex numbers and their behaviors in various mathematical applications.

Trigonometric Form of Complex Numbers

The trigonometric form of a complex number, also known as polar form, is another way to represent complex numbers involving a radius and an angle. It is written as \( r(\cos \theta + i \sin \theta) \), with \( r \) being the modulus or absolute value of the complex number, and \( \theta \) the angle, or argument, made with the positive real axis.

Conversion from trigonometric to Cartesian form involves trigonometric functions. The modulus \( r \) is the distance from the origin to the point, and the angle \( \theta \) specifies the direction. In our original exercise, we start with the trigonometric form and use the radius and angle to find the Cartesian form. Understanding how to navigate between these forms enhances flexibility in solving and interpreting complex numbers in different contexts.