Find the square root of 5-12i
To find the square root of 5 - 12i, we can use the formula for the square root of a complex number.
Let's express 5 - 12i in polar form.
The modulus or absolute value of a complex number is given by:
|r| = √(a^2 + b^2), where a is the real part and b is the imaginary part.
|r| = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13
The argument or angle of a complex number is given by:
θ = arctan(b/a), where a is the real part and b is the imaginary part.
θ = arctan(-12/5) ≈ -67.38°
Thus, 5 - 12i can be expressed in polar form as 13(cos(-67.38°) + sin(-67.38°)).
Now, for the square root of this number, we need to find another complex number whose square is equal to this number.
The square root formula for a complex number in polar form is:
√(r(cosθ + sinθ)) = √r(cos(θ/2) + sin(θ/2))
Applying this formula, we have:
√(13(cos(-67.38°) + sin(-67.38°))) = √13(cos(-67.38°/2) + sin(-67.38°/2))
√(13(cos(-67.38°) + sin(-67.38°))) ≈ √13(cos(-67.38°/2) + sin(-67.38°/2))
Using a calculator, we can evaluate this approximation:
√(13(cos(-67.38°/2) + sin(-67.38°/2))) ≈ √13(0.686 + 0.728i)
Therefore, the square root of 5 - 12i is approximately 0.686 + 0.728i.