## Atkinsoha 2 years ago PreCalc & Trig Help! Simplify the expression using a sum or difference formula: sin(x+y)sec x sec y

1. Atkinsoha

^opps, show all steps too please.

2. satellite73

$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$ then multiply out

3. Atkinsoha

how do you get to that point? like, can you start from the beginning because I need to show all steps for full credit.

4. satellite73

you can look in your text for a proof of that result, but that is not what you are being asked to do. you are being asked to multiply out, the point of the exercise it to use that identity, not to derive it

5. satellite73

in other words, you are being asked a question to see if you know the identity multiplying out is now a matter of algebra

6. Atkinsoha

I just don't understand how you went from sin(x+y)sec(x)sec(y) to that..

7. satellite73

oh i have confused you that is not the answer at all that is the identity $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$

8. Atkinsoha

Yes, you have confused me..

9. satellite73

now your job is to multiply $\left(\sin(x)\cos(y)+\cos(x)\sin(y)\right)\times \sec(x)\sec(y)$\]

10. Atkinsoha

Maybe I should tell you this.. I've been gone from school for a few weeks now (death in the family) I have no idea how to do any of this..

11. satellite73

you get $\sin(x)\cos(y)\sec(x)\sec(y)+\cos(x)\sin(y)\sec(x)\sec(y)$ as a first step

12. Atkinsoha

okay.. then were the heck do you go?

13. satellite73

that was multiplying out using the distributive law then $$\sec(x)=\frac{1}{\cos(x)}$$ so you can cancel some stuff

14. satellite73

lets look at the first term $\sin(x)\cos(y)\sec(x)\sec(y)$ that is equal to $\frac{\sin(x)\cos(y)}{\cos(x)\cos(y)}$

15. satellite73

cancel the $$\cos(y)$$ top and bottom and get $\frac{\sin(x)}{\cos(x)}$ which you can rewrite as $\tan(x)$

16. Atkinsoha

okay. that makes some sense..

17. Atkinsoha

basically the same thing for the other side too then?

18. satellite73

did you know $$\sec(x)=\frac{1}{\cos(x)}$$? this is really algebra at this step

19. satellite73

yes, same thing for the second term

20. satellite73

only this time you will get $$\tan(y)$$

21. Atkinsoha

Yes I did. Just didn't understand the problem.. so you get down to tan x=tanx?

22. Atkinsoha

ohh, yeah. tan x + tan y

23. satellite73

that is it, yes

24. Atkinsoha

25. satellite73

26. Atkinsoha

Okay, that makes sense. Do you have time to help me with one more problem?

27. satellite73

sure

28. Atkinsoha

okay, let me take a screen shot.

29. satellite73

k

30. Atkinsoha

31. Atkinsoha

I kind of understand it.. but not really :/

32. satellite73

ok we can do this, but it is going to take a minute ready?

33. Atkinsoha

Yes.

34. satellite73

we are going to use the same formula that we used above $\sin(u+v)=\sin(u)\cos(v)+\cos(u)\sin(v)$ at the moment, we know only two of those four numbers

35. Atkinsoha

yes, we know sin u and cos v

36. satellite73

we need $$cos(u)$$ do you know how to find it?

37. Atkinsoha

Don't you have to make a triangle or something?

38. satellite73

yes exactly

39. Atkinsoha

Okay, I'm not sure exactly how to come about finding it on a triangle

40. satellite73

|dw:1362154017344:dw|

41. satellite73

third side via pythagoras $\sqrt{61^2-11^2}=60$

42. Atkinsoha

Wouldn't it be 50?

43. satellite73

no $\cos(u)=\frac{60}{61}$

44. satellite73

45. Atkinsoha

Okay. so we still ned to find the sin one right?

46. satellite73

right but actually i made a mistake $\cos(u)=-\frac{60}{61}$ because you are in quadrant 2

47. satellite73

now we need $$\sin(v)$$ and it is the same idea as before |dw:1362154367573:dw|

48. satellite73

the third side is $\sqrt{41^2-40^2}=9$so $$\sin(v)=\frac{9}{41}$$

49. satellite73

now plug the numbers directly in to the formula

50. Atkinsoha

okay, then you just solve it right? multiplying and adding fractions basically?

51. satellite73

$\frac{11}{61}\times (-\frac{40}{41})+\frac{9}{61}\times(- \frac{60}{61})$

52. satellite73

yeah it is arithmetic from here on in gotta run, good luck

53. Atkinsoha

thank you!

54. Atkinsoha

wait quick

55. satellite73

56. Atkinsoha

for multiplying you flip the second fraction?

57. satellite73

$\frac{11}{61}\times (-\frac{40}{41})+\frac{9}{41}\times(- \frac{60}{61})$

58. satellite73

oh no

59. satellite73

flip nothing multiply straight across

60. Atkinsoha

okay. ill give it a try... thanks!

61. satellite73

yw