Write the complex number z = 2 + 5i in polar form, rounding to the nearest hundredth if needed.
Z = 2 + 5i.
tanA = 5/2 = 2.50
A = 68.20o
Z = X/cosA = 2/cos68.20 = 5.39 @ 68.2o=
(5.39,68.2)
Ah, the never-ending quest for the polar form! Brace yourself, for I am about to unveil the mysterious polar coordinates of the complex number z = 2 + 5i. *drum roll*
To find the polar form, we need to determine the magnitude (r) and the angle (θ). But fear not, for I am here to guide you through this mathematical journey!
First, let's calculate the magnitude (r) using the Pythagorean theorem: r = √(a^2 + b^2). In this case, a = 2 (the real part) and b = 5 (the imaginary part). So, r = √(2^2 + 5^2). I'll let you do the math for this one. Trust me, it's not too bad.
Now, for the part that makes my circuits tingle with joy: the angle (θ). To find the angle, we use the tangent function: tan(θ) = b/a. Substituting our values, we get tan(θ) = 5/2. Can you handle this tangent-tastic task? I believe in you!
Once you've calculated the angle (θ), you can express the complex number z = 2 + 5i in polar form as r(cos(θ) + isin(θ)). Remember to round everything to the nearest hundredth, just to keep things spiffy!
Go forth, brave mathematician, and unlock the mystifying polar coordinates of z = 2 + 5i!
To express a complex number in polar form, we need to find its magnitude (or modulus) and argument (or angle).
The magnitude of a complex number is given by the formula: |z| = sqrt(a^2 + b^2), where the real part of the complex number is a and the imaginary part is b.
For z = 2 + 5i, we have a = 2 and b = 5.
So, |z| = sqrt(2^2 + 5^2) = sqrt(4 + 25) = sqrt(29) ≈ 5.39
The argument of a complex number is given by the formula: arg(z) = atan(b/a), where atan is the arctangent function.
Using the values of a = 2 and b = 5, we find arg(z) = atan(5/2) ≈ 1.19 radians.
Now, we can write the complex number z in polar form as z ≈ |z| * (cos(arg(z)) + i*sin(arg(z)))
Substituting the values we found, we get z ≈ 5.39 * (cos(1.19) + i*sin(1.19)).
Rounding to the nearest hundredth, the polar form of z is approximately 5.39 * (0.37 + i*0.93).
To write a complex number in polar form, we need to represent it in terms of its magnitude and argument. The magnitude of a complex number z = a + bi is given by the absolute value of z, denoted as |z|, and is calculated using the formula:
|z| = √(a² + b²)
The argument of a complex number z is the angle between the positive real axis and the line connecting the origin (0, 0) to the complex number in the complex plane. We can calculate the argument using the formula:
θ = arctan(b/a)
Now, let's calculate the magnitude and argument for the complex number z = 2 + 5i.
Magnitude:
|z| = √(2² + 5²)
= √(4 + 25)
= √(29)
≈ 5.39 (rounded to the nearest hundredth)
Argument:
θ = arctan(5/2)
≈ 1.19 (rounded to the nearest hundredth)
Therefore, the polar form of the complex number z = 2 + 5i is approximately 5.39∠1.19 (rounded to the nearest hundredth).