## anonymous one year ago Help for a medal? I'm really struggling with limits. (I'll post the problem below)

1. anonymous

2. jim_thompson5910

Did you make a random guess at 1/4 ?

3. anonymous

Yes, I know that it was the right answer because the system told me it was. Luckily I wasn't graded on it.

4. anonymous

I just tried every option until it said correct, since all my answers were wrong.

5. jim_thompson5910

I see

6. jim_thompson5910

|dw:1441858889495:dw|

7. jim_thompson5910

what is sin^2 equal to in terms of cos^2 ?

8. anonymous

I personally don't have a clue. I know that cos(0) = 1 and sin(0) = 0 but that's about it

9. jim_thompson5910

you've learned that cos^2 + sin^2 = 1 right?

10. anonymous

I probably have in the past.

11. jim_thompson5910

look on page 2 of the pdf http://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf look at the pythagorean identities

12. jim_thompson5910

look familiar?

13. anonymous

Oh wow! I do recognize a lot of this stuff. Thanks for the pdf.

14. jim_thompson5910

yeah it's handy to keep it as a reference. Print it out if you can. Anyways, solving for sin^2 gives $\Large \sin^2(\theta) = 1-\cos^2(\theta)$

15. jim_thompson5910

|dw:1441859333706:dw| I'm getting everything in terms of cos(theta)

16. jim_thompson5910

what would $$\Large 1-\cos^2(\theta)$$ factor to?

17. anonymous

$\sin^2(\theta)$ ??

18. jim_thompson5910

hint: think of a^2 - b^2

19. anonymous

1?

20. jim_thompson5910

another hint: a^2 - b^2 = (a-b)(a+b)

21. anonymous

I'm sorry, but I am completely confused. Math used to be my best subject, though that doesn't seem to be the case anymore...

22. jim_thompson5910

if you had x^2 - 5^2, what would that factor to?

23. anonymous

(x+5)(x-5)

24. jim_thompson5910

so like that idea, we factor 1 - cos^2 into (1+cos)*(1-cos)

25. jim_thompson5910

|dw:1441860277846:dw|

26. anonymous

Oh okay. I see now.

27. jim_thompson5910

I'm sure you see what cancels?

28. anonymous

Yes the two (1-cos theta) ones cancel

29. jim_thompson5910

yep |dw:1441860261204:dw|

30. jim_thompson5910

we're now left with $\Large \frac{1}{2(1+\cos(\theta))}$

31. jim_thompson5910

at this point, plugging in theta = 0 does not lead to a division by zero error (as it did before in the previous expressions). So you can now plug in theta = 0 and simplify

32. anonymous

then it would be... 1 / 2(1+1) 1 / 4

33. jim_thompson5910

yep

34. anonymous

You're amazing! Thank you so much!

35. jim_thompson5910

no problem