Expert: Alon Mandes - 4/30/2009

Question
anything you can answer REALLY DOES HELP!

So first part is:

1. to this trig identity, sin(x+h)=sin(x)cos(h)+cos(x)sin(h)..
by using factoring and limit laws how do I simplify this further so that there is no x's in the limits??

2. What is the area of a sector with angle Pi/2 and radius r?

3. Find a formula for the area of a sector with angle (theta) and radius r.
Area of sector= (area of circle)(fraction of circle)=

4. from this picture,(it looks like a triangle with a rounded part at one side as taken from a circle)with theta as the angle and r on both sides. theta is facing the curved circle edge. The question is:

Area of small triangle < area of circle < Area of large triangle

Substitue formulas for the areas into this inequality.

5. After I find answer to 4, it says I need to multiply by 2 to simplify the inequality.

6. says I need to substitute i think t=(cos(theta),sin(theta)) in the inequality to part 5 above.

7. it then says if a<b, what's the relationship between 1/a and 1/b, I said 1/a > 1/b.. is that suppose to be right?

8. use part 6 above to invert the inequality in part f.

9. multiply by sin(theta). you should now have sin(theta)/(theta) between two other quantities.

10. Use the squeeze theorem to find lim    sin(theta)
                               (theta)->0 ---------
                                           (theta)


theres more but I don't want to overload my questions cuz I know its too much to ask but anything you can answer would help me so much. Its a step by step thing i'm suppose to do and if i can't get one part I can't get another part later on that refers to it. So anything really does help.

thank you!  

Answer
1. Lim  [Sin(x+h)-Sin(x)]/h =
  h->0

Lim [sin(x)cos(h)+cos(x)sin(h)-sin(x)]/h =
h->0

Lim { sin(x)[cos(h)-1] + cos(x)sin(h) } / h =
h->0

Lim   sin(x)[cos(h)-1]/h    +    Lim    cos(x)sin(h)/h
h->0                             h->0

1st part = 0                &     2nd part = cos(x)
Proving that will be in section 8,9 & 10.

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(2+3). What is the area of a sector with angle Pi/2 and radius r?
& Find a formula for the area of a sector with angle (theta) and radius r .
The 2 major rules are :
α/2π=L/2πr  -->  L=αr . Where α in radians & L is the arc .
α/2π=S/πr²  -->  S=½αr² . Where α in radians & S area of sector .


Alon.