De formule van Euler

Antwoorden bij de opgaven

1. 1. Of je zo mag differentiëren is nog maar de vraag; daarvoor moet je eerst meer theorie opbouwen!
   2. `z=r(cos(phi)+isin(phi))=r*text(e)^(text(i)phi)`
   3. `z=sqrt(8)*text(e)^(-0,25pitext(i))`
2. 1. `z_1=sqrt(2)*text(e)^(-0,25pi text(i))`
   2. `z_2=sqrt(2)*text(e)^(0,75pi text(i))`
   3. `z_1*z_2=2*text(e)^(0,5pi text(i))`
   4. `z_1*z_2=2text(i)`
   5. Duidelijk, toch?
3. 1. `z=3text(e)^text(i)`
   2. `z~~-1,980+0,282text(i)`
4. 1. -
   2. -
   3. `z~~5text(e)^(-0,93text(i))`
5. 1. -
   2. `|z|=sqrt(20)` en `arg(z)~~2,68`
   3. `z=sqrt(20)*text(e)^(2,86text(i))`
   4. -
6. 1. -
   2. -
   3. -
7. 1. -
   2. `-4-4text(i)`
   3. klopt natuurlijk
   4. -
8. 1. -
   2. `(text(e)^(text(i)phi))^n=text(e)^(text(i) n phi)`
9. 1. `z=2text(e)^(0text(i))`
   2. `z=1text(e)^(0,5pitext(i))`
   3. `z=3text(e)^(0,5pitext(i))`
   4. `z=sqrt(2)*text(e)^(-0,25pitext(i))`
   5. `z=sqrt(2)*text(e)^(0,75pitext(i))`
   6. `z=sqrt(8)*text(e)^(1,25pitext(i))`
10. `z=sqrt(2)*text(e)^(5/12 pi)`
11. `z_1=2i; z_2=cos(1)+text(i)sin(1); z_3=-3+text(i)sqrt(3); z_4=-1; z_5=1`
12. 1. `-128 + 128text(i)`
    2. `122+597text(i)` (niet afronden tussentijds!)
13. `z_1=-0,5; z_2~~2,17-0,58text(i)`
14. 1. -
    2. `text(e)^(text(i)phi)+text(e)^(-text(i)phi)=2cos(phi)`
    3. `sin(phi)=-0,5i(text(e)^(text(i)phi)+text(e)^(-text(i)phi))`
15. 1. `z=5+3text(i)`
    2. `z=3/34 + 5/34 text(i)`
    3. `z=738-684text(i)`
    4. `z=-0,5 + 0,5text(i)sqrt(3)`