SOLUTION: Hello, how do I figure out this complex numbers problem? Any help is greatly appreciated! The point (-4+bi)has been plotted on the complex plane and r=8. If 0°≤θ&#880

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Hello, how do I figure out this complex numbers problem? Any help is greatly appreciated! The point (-4+bi)has been plotted on the complex plane and r=8. If 0°≤θ&#880      Log On


   

Question 396764: Hello, how do I figure out this complex numbers problem? Any help is greatly appreciated!
The point (-4+bi)has been plotted on the complex plane and r=8. If 0°≤θ≤180° find the value of b. Leave answer as a radical.
Thank you.
Found 2 solutions by jsmallt9, richard1234:
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The "a" and "b" in standard form and the "r" in polar form are related by the equation:
a%5E2+%2B+b%5E2+=+r%5E2
Subsituting the value we have for "a" and "r" we get:
%28-4%29%5E2+%2B+b%5E2+=+%288%29%5E2
Simplifying we get:
16+%2B+b%5E2+=+64
Subtracting 16 we get:
b%5E2+=+48
Now we find the square root of each side. Since the theta is between 0 and 180, we want "b" to be positive. (A negative "b" would put theta between 180 and 360.) So we will only use the positive square root:
b+=+sqrt%2848%29
All that is left is simplifying:
b+=+sqrt%2816%2A3%29
b+=+sqrt%2816%29%2Asqrt%283%29
b+=+4%2Asqrt%283%29

Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
The magnitude of the complex number is sqrt%28%28-4%29%5E2+%2B+b%5E2%29+=+8. Simplifying,

16+%2B+b%5E2+=+64

b%5E2+=+48

b+=+0+%2B-+sqrt%2848%29+=+0+%2B-+4sqrt%283%29. Note that b must be positive since the angle is between 0 and 180, so b+=+4sqrt%283%29.