Help solving complex number Math problem

In summary, the conversation was about solving a question related to a complex number and its representation on an Argand diagram. The complex number z is given by z = 1 + cos (&theta;) + i sin (&theta), where -pi < theta < or = +pi. It was shown that for all values of theta, the point representing z in the Argand diagram is located on a circle and the center and radius of the circle can be found using parametric equations. The conversation then moved on to discussing the real part of 1/z, which was proven to be 1/2 for all values of theta.
  • #1
denian
641
0
hope to get the idea on how to solve this question.

the complex number z is given by

z = 1 + cos (theta) + i sin (theta)

where -pi < theta < or = +pi


show that for all values of theta, the point representing z in a Argand diagram is located on a circle. find the centre and radius of the circle.
 
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  • #2
If z = 1 + cos (&theta;) + i sin (&theta)

Then z-1= cos(&theta)+ i sin(&theta;).

If you represent z as x+ iy then
(x-1)+ iy= cos(&theta;)+ i sin(&theta;)

or x- 1= cos(&theta;), y= sin(&theta;)

Those are parametric equations of a circle with what center and radius?
 
  • #3
ok. i compare those with the
y=r sin (&theta)
and
x=r cos (&theta)


so, i know the radius = 1 unit
but may i know how to find the centre of the circle?

tq.
 
  • #4
It's exactly where the center of the circle given by

x= r cos &theta;
y= r sin &theta; is!

Hint: x2= r2cos2&theta;
y2= r2sin2&theta;

What is x2+ y2?

If that's too complicated, what is (x,y) when &theta;= 0?
What is (x,y) when &theta;= &pi;?
 
  • #5
tq. u helped me solved the problem.

but there is another part of the question which i ahain need some idea.

--> prove that the real part of (1/z) is (1/2) for all values of [the]
 

1. What are complex numbers?

Complex numbers are numbers that consist of both a real and imaginary part. The real part is a regular number, while the imaginary part is a multiple of the imaginary unit, designated by the letter i.

2. How do I add or subtract complex numbers?

To add or subtract complex numbers, simply combine the real parts and the imaginary parts separately. For example, (3 + 5i) + (2 + 4i) = (3 + 2) + (5i + 4i) = 5 + 9i.

3. How do I multiply complex numbers?

To multiply complex numbers, use the distributive property and FOIL method. For example, (3 + 5i) * (2 + 4i) = 3 * 2 + 3 * 4i + 5i * 2 + 5i * 4i = 6 + 12i + 10i + 20i^2 = -14 + 22i.

4. How do I divide complex numbers?

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number is the same number with the sign of the imaginary part changed. For example, (3 + 5i) / (2 + 4i) = ((3 + 5i) * (2 - 4i)) / ((2 + 4i) * (2 - 4i)) = (6 + 10i - 12i + 20i^2) / (4 + 8i - 8i - 16i^2) = (-14 + 22i) / 20 = -0.7 + 1.1i.

5. How do I solve complex number equations?

To solve complex number equations, treat them like regular equations and use algebraic techniques such as combining like terms, isolating the variable, and using the appropriate operations to find the solution. Remember to apply the rules of complex numbers when performing operations.

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