## vishweshshrimali5 Group Title Solve: $\large{\theta = \tan^{-1}(2\tan ^2\theta) - \frac{1}{2} \sin^{-1} \frac{3 \sin 2 \theta }{ 5 + 4 \cos 2 \theta }}$ one year ago one year ago

1. vishweshshrimali5 Group Title

@hartnn @amistre64 @Callisto @Hero

2. amistre64 Group Title

and what is it we are looking to do with this monstrocity?

3. vishweshshrimali5 Group Title

Solve for theta

4. vishweshshrimali5 Group Title

(general solutions)

5. amistre64 Group Title

im just too sick to be able to concentrate on most of that :/

6. vishweshshrimali5 Group Title

Its OK

7. vishweshshrimali5 Group Title

Eventhough Thanks for your time

8. amistre64 Group Title

${\theta = \tan^{-1}(2\tan ^2\theta) - \frac{1}{2} \sin^{-1} \frac{3 \sin 2 \theta }{ 5 + 4 \cos 2 \theta }}$ ${t = \tan^{-1}(2sec^2t-2) - \frac{1}{2} \sin^{-1} \frac{3 \sin 2t }{ 5 + 4 \cos 2t }}$ ${2t = 2\tan^{-1}(2sec^2t-2) - \sin^{-1} \frac{3 \sin 2t }{ 5 + 4 \cos 2t }}$ yeah, im not going to be able to work it out in this condition :) good luck tho

9. hartnn Group Title

maybe use $$arcsin x= 2arctan (x/(1+\sqrt{1-x^2}))$$

10. harsh314 Group Title

$\frac{ 1 }{ 2 }\sin^{-1} \frac{ 3\sin \theta }{ 5+4 \cos \theta }$be assumed as $\alpha$ now$\sin 2\alpha =\frac{ 3 \sin 2 \theta }{ 5+4\cos2 \theta }=m$ suppose now$m=\frac{ 6 \tan \theta }{ 9+\tan ^{2}\theta } ,$

11. harsh314 Group Title

$\sqrt{\frac{ 1-m }{ 1+m }}=\tan{\frac{ \pi }{ 4 } -\alpha}$ $\sqrt{\frac{ 1-m }{ 1-m }}=\frac{ 3-\tan \theta }{ 3+\tan \theta }$ $\frac{ \pi }{ 4 }- \alpha=\tan^{-1} \sqrt{\frac{ 1-m }{ 1+m }}$ substituting these values in the equation we obtain a cubic equation$\tan ^{3}\theta -3\tan \theta+2=0$ i don't know how to solve it plz kindly tell me still by hit and trial i got pi/4

12. hartnn Group Title

to solve, x^3-3x+2=0 i would first note that 1-3+2 =0 hence (x-1) must be the factor, x=1 must be the root tan theta =1 gives theta = pi/4

13. hartnn Group Title

to get other 2 roots, we can use synthetic division, taking x=1

14. harsh314 Group Title

other values are tan theta =2,-1 ,thanks hartnn