## Loser66 one year ago Check my stuff, please find z : cos z = 3i

1. Loser66

$$cos z = \dfrac{e^{iz}+e^{-iz}}{2}= 3i$$ $$e^{iz}+ e^{-iz} -6i =0$$ $$e^{2iz}-6ie^{iz} +1=0$$ Let t = e^(iz)

2. Loser66

$$t^2 -6it +1=0\\t = 3i \pm i\sqrt{10}$$

3. misty1212

HI!!

4. misty1212

this looks good, solving a quadratic i think there is another way too, but this should work

5. misty1212

let me try a different way

6. misty1212

oh

7. misty1212

i still want to try a different way

8. misty1212

$\cos(z)=3i\\ \cos^2(z)=-9\\ 1-\sin^2(z)=-9\\ \sin^2(z)=10\\ \sin(z)=\sqrt{10}$

9. misty1212

ok lets skip this maybe i can't do the other one, not sure

10. Loser66

$$t = i(3+\sqrt {10})= e^{iz} \\ iz = log(i(3+\sqrt{10}) = log| (i(3+\sqrt{10})| + i (arg (i(3+\sqrt{10}) +2k\pi ~~~k\in \mathbb Z$$

11. Loser66

but $$|i(3+ \sqrt{10}| = 3 + \sqrt{10}$$ and $$3+\sqrt{10}>0$$ hence $$arg (i(3+\sqrt{10}) = \pi/2$$ that gives us $$z = \{(pi/2 + 2k\pi) -i log(3+\sqrt{10}\}$$

12. Loser66

For $$t = i(3-\sqrt{10})$$. since $$3-\sqrt{10} <0$$ its argument is -\pi/2 the same process but replace the arg, we have $$z = \{(pi/2 + 2k\pi) -i log(3+\sqrt{10}\} \cup \{(-\pi/2 + 2k\pi) -i log(3-\sqrt{10}\}$$

13. Loser66

@ganeshie8

14. anonymous

couldn't you just do $z=\cos ^{-1}(3i)+2k \pi$

15. Loser66

if it is so, cos^-1 (3i) gives me just one case of the angle of 3i, not the real part

16. anonymous

yeah i get you

17. anonymous

18. anonymous

brb gimmi 2mins gotta do something

19. anonymous

you look like your on the right track

20. anonymous

$z=2k \pi -iln \left[ i(3\pm \sqrt{10}) \right] k \epsilon R$

21. anonymous

mmm

22. anonymous

would that suffice?

23. Loser66

how?

24. Loser66

|dw:1444384172324:dw|

25. anonymous

i think you're alright

26. anonymous

omg, someone take over, i have dinner calling >.< sorry

27. Loser66

I'm ok. Thanks chris00

28. mom.

can we do something with logarithms? lets take log on both sides we get- $\log_{e}cosz =\log_{e}3i$ $\log_{e}3i=\log_{e}(3)+i \tan^{-1} 3=log_{e}cosz$

29. Loser66

We need z =..... , not log (3i) Such a nick. hahaha... whenever I call you, your nick makes me miss my death Mom.