Graph each complex number. Then express it in rectangular form.
2(cos(4pi/3) + isin(4pi/3))
I don't understand where to go from here.
cos(4pi/3) = - 1/2
sin(4pi/3) = - √3/2
so
2(cos(4pi/3) + isin(4pi/3)) = -1/2 - (√3/2)i
here is a nice page for this topic
http://www.allaboutcircuits.com/vol_2/chpt_2/5.html
To graph the complex number 2(cos(4pi/3) + isin(4pi/3)), we can convert it to rectangular form and then plot it on the complex plane.
In trigonometric form, a complex number can be represented as r(cosθ + isinθ), where r is the magnitude (or modulus) of the complex number and θ is the argument (or angle) in radians.
In this case, r = 2 and θ = 4pi/3. Plugging these values into the formula, we have:
2(cos(4pi/3) + isin(4pi/3)) = 2 * [(cos(4pi/3) + isin(4pi/3))]
To convert it to rectangular form, we can simplify the expression inside the brackets using Euler's formula, which states that e^(iθ) = cosθ + isinθ:
2 * [(cos(4pi/3) + isin(4pi/3))]
= 2 * (e^(i*4pi/3))
= 2 * (cos(4pi/3) + isin(4pi/3))
Expanding further:
= 2 * (cos(4pi/3) + i*sin(4pi/3))
= 2 * (-1/2 + i*(-√3/2))
= -1 + i*(-√3)
Therefore, the rectangular form of the complex number 2(cos(4pi/3) + isin(4pi/3)) is -1 + i*(-√3).
Now we can plot it on the complex plane. The real part is -1 and the imaginary part is -√3. So the graph of the complex number is the point (-1, -√3) or (-1, -1.732).
To graph the complex number, you need to plot it on the complex plane. The complex plane consists of two axes: the real axis (horizontal) and the imaginary axis (vertical). Complex numbers are represented as points on this plane.
To express the given complex number in rectangular form, you need to convert it from polar form. Rectangular form, also known as Cartesian form, represents a complex number as a combination of a real part and an imaginary part.
Let's break down the given complex number: 2(cos(4π/3) + isin(4π/3)).
Step 1: Plot the complex number on the complex plane.
To graph the complex number, start at the origin (0,0). Since the magnitude is 2, we move two units away from the origin in the direction of the angle 4π/3. This can be visualized as starting at the positive x-axis and moving 2 units counter-clockwise (since 4π/3 is in the second quadrant).
Step 2: Determine the rectangular form.
In polar form, a complex number can be represented as z = r(cosθ + isinθ), where r is the magnitude and θ is the angle. To convert it to rectangular form, we can use the trigonometric identities that relate cosine and sine to real and imaginary parts.
In this case, r = 2 and θ = 4π/3.
cos(4π/3) = -1/2
sin(4π/3) = -√3/2
Substituting these values into the rectangular form, we have:
z = 2(-1/2 + i(-√3/2))
z = -1 - √3i
So the complex number 2(cos(4π/3) + isin(4π/3)) can be expressed in rectangular form as -1 - √3i.
Note: The rectangular form represents the real part as -1 and the imaginary part as -√3. The negative signs indicate direction and the square root term √3 represents the magnitude.