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.
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?