# Math Help - trig applications # 3

1. ## trig applications # 3

Points A and B lie on a circle, centre O, radius 5 cm. Find the value of angle AOB that produces a maximum area for the triangle AOB.

This is my work and attempt:

So I will describe my diagram. It is a circle, with a triangle inside (not right angled) in the middle of the circle is a dot labelled O. From here I randomly drew point A and point B, on the edge of the circle and then connected points AOB to form a triangle in the circle. Side OB will be labelled 5 cm as well as AO for radius.

Then in the middle of AB i chopped it in half and drew a line down the middle to point O to form two right angled triangles.

Now I can begin:
cosx=a/h
cosx=a/5
5cosx=a

sinx=o/h
sinx=o/5
5sinx=o

Max area of triangle
A=0.5 b h
=(0.5) 5cosx (5sinx)
=12.5(cosxsinx)

A'=-12.5(sin^(2)x-cos^(2)x)

A'=0 when x=pi/4
But this is only half of theta or angle O, so times two.
pi/4 x 2
=pi/2

Now this answer is correct with the back of the book BUT It doesn't work so well because when i plug it back into my area formula, it turns out the angle is 0.. which cannot be max?

A(0)=0
A(pi/2)=0
So my area formula must be wrong.

What is wrong from my work and how do i fix it? demonstrate.

2. $A = \frac{1}{2}r^2\sin{\theta}$

3. Originally Posted by skeeter
$A = \frac{1}{2}r^2\sin{\theta}$
how did you get the base and height to be that?

4. Originally Posted by skeske1234
how did you get the base and height to be that?
Math B - Area of Triangle Using Trigonometry

5. shouldn't it be
A=1/2 (r^2) (5sinx)?
How did you just get sinx using the SAS rule?
like even if you use the SAS rule, that means that you know two sides of a triangle and an angle inbetween.. i know that, but my question is how is the formula for that derived (sinx) to represent the height of the triangle?
and another question, r^2 represents the diamater correct?

6. Originally Posted by skeske1234
shouldn't it be
A=1/2 (r^2) (5sinx)?
How did you just get sinx using the SAS rule?
like even if you use the SAS rule, that means that you know two sides of a triangle and an angle inbetween.. i know that, but my question is how is the formula for that derived (sinx) to represent the height of the triangle?
and another question, r^2 represents the diamater correct?