Write the complex number 6-6i in polar form with argument θ between 0 and 2π.
6-6i
magnitude = √(36 + 36) = √72 = 6√2
angle:
tanθ = -6/6 = -1 , where θ is in 4th quadrant of the Argand plane
θ = 315° or 7π/4 radians
6 - 6i = 6√2(cos(7π/4) + i sin(7π/4))
sometimes shortened to 6√2 cis (7π/4)
Well, well, well, look who we have here, a complex number in need of some clowning around! Time to put on my polka-dot hat and dive into the polar form.
To convert a complex number into polar form, we need to find its magnitude (r) and argument (θ). The magnitude can be found using the good old Pythagorean theorem: r = √(a^2 + b^2), where a is the real part and b is the imaginary part.
In this case, a = 6 and b = -6i. So, let's calculate the magnitude first:
r = √(6^2 + (-6)^2) = √(36 + 36) = √72 = 6√2
Next, we need to determine the argument (θ). Hmmm, let's consider the point on the complex plane that this number represents. Since the imaginary part is negative (-6i), it's located in the fourth quadrant.
But wait, we're given a range for the argument θ between 0 and 2π, which covers the entire complex plane! So, let's find an angle in the fourth quadrant that gives us the same sine and cosine values as our complex number.
Since sine is negative in the fourth quadrant, and cosine is positive, we can conclude that θ = -π/4. But hold on a sec, we need to get our angle between 0 and 2π. And how do we do that? We add 2π, of course!
So, θ = -π/4 + 2π = 7π/4
Putting it all together, the complex number 6 - 6i in polar form, with the argument θ between 0 and 2π, is:
6√2 * e^(i * 7π/4)
And there you have it, a wacky polar form for our complex number. Enjoy the circus ride through mathematics, my friend! 🎪😄
To write the complex number 6-6i in polar form, we need to calculate its magnitude (r) and argument (θ).
The magnitude (r) of a complex number is given by the formula:
r = √(a^2 + b^2)
For the complex number 6-6i, the real part (a) is 6 and the imaginary part (b) is -6. Plugging in the values, we have:
r = √(6^2 + (-6)^2) = √(36 + 36) = √72 = 6√2
The argument (θ) can be calculated using the formula:
θ = arctan(b/a)
Plugging in the values, we have:
θ = arctan((-6)/6) = arctan(-1) = -π/4
Since the argument should be between 0 and 2π, we can adjust the value of θ by adding 2π:
θ = -π/4 + 2π = (7π)/4
Therefore, the complex number 6-6i in polar form with an argument θ between 0 and 2π is:
6√2 * e^((7πi)/4)
To express a complex number in polar form, we need to find the modulus (r) and the argument (θ) of the complex number. The modulus represents the distance of the complex number from the origin, and the argument represents the angle the complex number makes with the positive real axis.
To find the modulus (r), we can use the formula:
r = √(a^2 + b^2)
Where 'a' is the real part of the complex number, and 'b' is the imaginary part.
In this case, a = 6 and b = -6i. So, the modulus (r) will be calculated as:
r = √(6^2 + (-6)^2) = √(36 + 36) = √72 = 6√2
To find the argument (θ), we can use the formula:
θ = tan^(-1)(b/a)
In this case, a = 6 and b = -6. So, the argument (θ) will be calculated as:
θ = tan^(-1)(-6/6) = tan^(-1)(-1) = -π/4
However, we want the argument (θ) to be between 0 and 2π. The argument we obtained (-π/4) is in the fourth quadrant, but we can add 2π to get it into the desired range.
Adding 2π to the argument:
θ = -π/4 + 2π = 7π/4
Now we have both the modulus (r) and the argument (θ). So, we can express the complex number 6 - 6i in polar form as:
6 - 6i = 6√2 * (cos(7π/4) + i*sin(7π/4))