how to express 3+4i in polar form
To express a complex number in polar form, you need to find its magnitude (r) and argument (θ).
To express 3+4i in polar form, follow these steps:
Step 1: Calculate the magnitude (r)
The magnitude (r) of a complex number is the distance from the origin to the point representing the complex number in the complex plane. It can be calculated using the formula:
r = √(a^2 + b^2)
In this case, a = 3 (the real part) and b = 4 (the imaginary part). Thus, we have:
r = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
So, the magnitude (r) of 3+4i is 5.
Step 2: Calculate the argument (θ)
The argument (θ) of a complex number is the angle between the positive real axis and the line connecting the origin and the complex number in the complex plane. It can be calculated using the formula:
θ = arctan(b/a), where arctan is the inverse tangent function.
In this case, a = 3 (the real part) and b = 4 (the imaginary part). Thus, we have:
θ = arctan(4/3)
Using a calculator or a math software, you can find the value of arctan(4/3) to be approximately 0.93 radians (or approximately 53.13 degrees).
So, the argument (θ) of 3+4i is approximately 0.93 radians (or approximately 53.13 degrees).
Step 3: Express the complex number in polar form
To express the complex number 3+4i in polar form, we write it as:
3+4i = 5(cos(θ) + isin(θ))
Since r = 5 and θ ≈ 0.93 radians (or ≈ 53.13 degrees), we can write 3+4i in polar form as:
3+4i = 5(cos(0.93) + isin(0.93))
Therefore, 3+4i in polar form is 5(cos(0.93) + isin(0.93)).