Express the complex number in polar form. 5-12i
usually they are of form:
a i +/- b j
OIC, ok
r = sqrt (25 + 144) = sqrt (169 ) = 13
angle in quadrant 4
tan angle below x axis = 12/5
angle below x axis = 67.4 degrees
or
360 - 67.4 = 292.6 degrees
r=13, θ≈-67.3801º
5 - 12i
r = 13 ,
but we want it in the form
r(cosØ + isinØ)
cosØ = 5/13 , so Ø could be in I or IV
sinØ = -12/13, so Ø could be in III or IV
so Ø must be in IV
taking Damon's angle of 292.6°
5 - 12i = 13(cos292.6 + i sin292.6) in polar form
check:
13cos296.6 = 4.9958.. or 5
13sin292.6 = -12.0017.. or -12
To express a complex number in polar form, we need to find its magnitude (r) and argument (θ). The magnitude of a complex number is the distance from the origin (0,0) to the point representing the complex number in the complex plane. The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the point representing the complex number.
To calculate the magnitude (r), we use the formula:
r = √(Re^2 + Im^2)
where Re is the real part of the complex number and Im is the imaginary part. In this case, the real part is 5 and the imaginary part is -12. Therefore,
r = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13
To find the argument (θ), we use the formula:
θ = atan(Im/Re)
where atan is the inverse tangent function. In this case, the real part is 5 and the imaginary part is -12. Therefore,
θ = atan((-12)/5) ≈ -1.176
Now, we have the magnitude (r) and argument (θ). The complex number 5-12i can be expressed in polar form as:
5-12i = 13(cos(-1.176) + i*sin(-1.176))
Note: The angle (-1.176) is in radians.