Argand diagram regions and complex modulus with trigonometric form

[fasterthanjoao]

1) Sketch the regions in the Argand diagram given by:

|z+i| < |z-3|

this means the distance from z to 3 is greater than the distance from z to i? would I then just draw two lines on the argand diagram, shading the lower region and not including the boundary? haven't had a problem with these questions through term, guess I am just a bit burned out and need to make sure.

2) Find the modulus and the argument of:
1+ cosx + isinx

does this involve changing the angle to x/2 and using trig expansions?

-thanks.

 

[marlon]

1) Sketch the regions in the Argand diagram given by:

|z+i| < |z-3|

this means the distance from z to 3 is greater than the distance from z to i? would I then just draw two lines on the argand diagram, shading the lower region and not including the boundary? haven't had a problem with these questions through term, guess I am just a bit burned out and need to make sure.

2) Find the modulus and the argument of:
1+ cosx + isinx

does this involve changing the angle to x/2 and using trig expansions?

-thanks.

In this case it is useful to express these complex numbers as z = a+ib and then calculate the modulus. You know that a is on the x-axis and y is on the imaginary y-axis...

This oughtta help you out...i hope

For example : Z + i = a + (b + 1)i

Write out the inequality and look at a and b as the x and y coordinates of some first order inequality that you will need to solve... I tried it like this and it worked
marlon

 

[wais]

1) To sketch the regions in the Argand diagram given by |z+i| < |z-3|, you would first need to understand what this inequality represents. In general, the inequality |z-a| < |z-b| means that the distance from z to a is less than the distance from z to b. So in this case, the distance from z to i is less than the distance from z to 3.

To sketch this on the Argand diagram, you would start by drawing the points i and 3. Then, you would draw a circle centered at i with a radius equal to the distance from i to 3. This represents all the points that satisfy the equation |z-i| = |z-3|. Next, you would shade the region inside this circle, as this represents all the points that satisfy the inequality |z-i| < |z-3|. Finally, you would draw the boundary of this shaded region, which would be the circle itself.

2) To find the modulus and argument of 1+cosx+isinx, you would first need to understand what these terms mean. The modulus of a complex number is its distance from the origin, which can be found using the Pythagorean theorem. In this case, the modulus would be √(1+cos^2x+sin^2x) = √2. The argument is the angle that the complex number makes with the positive real axis. To find this, you can use the inverse tangent function, which would give you an angle of x/2.

So in short, yes, you would need to use trigonometric expansions and identities to find the modulus and argument of this complex number.