Find the square root of the complex number
(a)5+12i
To find the square root of a complex number, we will use the square root property of complex numbers.
Let's start by expressing the complex number in the form a + bi, where a is the real part and b is the imaginary part.
For the complex number 5 + 12i, a = 5 and b = 12.
Next, we need to find the modulus (or magnitude) of the complex number, which can be calculated using the formula:
|z| = sqrt(a^2 + b^2)
In this case:
|5 + 12i| = sqrt(5^2 + 12^2)
= sqrt(25 + 144)
= sqrt(169)
= 13
Now, we can find the argument (or angle) of the complex number, which can be calculated using the formula:
arg(z) = atan(b/a)
In this case:
arg(5 + 12i) = atan(12/5)
≈ 1.176 radians (rounded to three decimal places)
Now, we can use the square root property of complex numbers:
The square root of a complex number z can be expressed as:
sqrt(z) = sqrt(r) * ( cos(theta/2) + i * sin(theta/2) )
Where r is the modulus and theta is the argument.
For our complex number 5 + 12i, the square root is:
sqrt(5 + 12i) = sqrt(13) * ( cos(1.176/2) + i * sin(1.176/2) )
Now we can calculate the square root using a calculator or by approximating the values of cos and sin.