Trigonometry with circle question

I am stuck with this problem. Any help?

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Since you are stuck in the beginning -

AC = AO + OB + BC = 3r

sin(x) = h/r

Area of triangle APC = 1/2 * b * h

Now continue....

Please share your work with us.

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I am stuck with this problem. Any help?
What is the formula for the area of a triangle?

What is the length of the base of triangle APC, regardless of its height?

How is the sine defined relative to the unit circle?

So what is the height of triangle APC in terms of r and x?

So what is the area of triangle APC?

What is the maximum value attainable by the sine function?

At what value of x does that occur?

So what is the maximum height attainable?

So what is its maximum area?
 
This is what i tried, I used the area rule instead,

Area of POC
1/2(2r)(3r)Sin(x)...........(OC is 2r and PC is 3r)
= 3r2Sinx

Area of AOP
1/2(r)(r)Sin(x)......(AO and PO is r)
= 1/2r2Sinx

Adding them gives me 7/2r2Sinx, What is wrong?
 
This is what i tried, I used the area rule instead,

Area of POC
1/2(2r)(3r)Sin(x)...........(OC is 2r and PC is 3r)
= 3r2Sinx

Area of AOP
1/2(r)(r)Sin(x)......(AO and PO is r)
= 1/2r2Sinx

Adding them gives me 7/2r2Sinx, What is wrong?
The base of triangle POC is b = OC = 2r
The height of triangle POC is OP sin(x), NOT PC sin(x). Thus h = r sin(X),
A = (1/2)bh = (1/2)(2r)(r sin(x))
EDIT: by "area rule," do you mean half the product of two sides times the sine of the angle between them?
.........In that case, the two sides bounding angle x are OP(=r) and OC (=2r).

Your area for triangle AOP is correct.

You could do the single triangle APC directly. Its base is b=3r and height h=r sin(x).
 
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Thanks, I see my mistake, but I am a bit confused with the next question and the question after that. Could you please show or explain it to me ?
 
Thanks, I see my mistake, but I am a bit confused with the next question and the question after that. Could you please show or explain it to me ?
What are your thoughts? What have you tried? Where are you stuck?

Please be complete. Thank you! ;)
 
For the second problem - JeffM gave you hints.

For the last problem, use law of cosines wrt triangle OPC.
 
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