mathslover 3 years ago find the general solution for : $$\sin^2 \theta -2 \cos \theta + \frac{1}{4} = 0$$

1. Yahoo!

$Sin^2\theta = 1-\cos^2 \theta$

2. Yahoo!

3. mathslover

I got $$\theta = \cos^{-1} \frac{1}{2}$$

4. mathslover

theta = 60 degrees.

5. mathslover

but general solution???

6. terenzreignz

Just add 2k(360 degrees) where k is any integer, and there's your general solution :D

7. Yahoo!

$\cos \theta =\frac{ 1 }{ 2 }$ $Cosx=Cosy$ then $x=(2n \Pi)\pm y$

8. terenzreignz

wait, no, k(360 degrees) not 2k sorry

9. calculusfunctions

May I explain?

10. mathslover

11. Yahoo!

$x= (2n \Pi) \pm \frac{ \Pi }{ 3 }$

12. mathslover

I had got this : cos theta = 1/2 , please explain after this.

13. Yahoo!

There Will be Two value For Cos x @mathslover

14. mathslover

Yes but the negative one will NOT BE ACCEPTED..

15. mathslover

as cos x can NOT BE NEGATIVE @Yahoo!

16. Yahoo!

Why min value of cos x = -1 ....

17. terenzreignz

Must x be in degrees?

18. terenzreignz

theta rather

19. Yahoo!

$\cos 180 = -1$

20. calculusfunctions

$\sin ^{2}\theta -2\cos \theta +\frac{ 1 }{ 4 }=0$First multiply the equation by the least common denominator.

21. mathslover

Wait, there will be two values for cos theta , the other one is -5/2 but -5/2 , but since the minimum value of cos theta is -1 and hence it can not be -5/2 is this what we must think abt @Yahoo! ? (sorry for my above explanation as : cos theta can never be negative as I thought that we are talking about cos(-theta) = cos theta)

22. mathslover

cos(theta) $$\ne$$ -5/2

23. Yahoo!

Yup.....nw u r Correct

24. calculusfunctions

@mathslover Let me know when you're interested in learning a more efficient method.

25. mathslover

@calculusfunctions sir, thanks a lot , I think I got it now.

26. mathslover

Sorry but do you have any easier or another way?

27. calculusfunctions

Yes if you're interested.