# An Applied Trig/Max and Min Problem

• Jan 10th 2012, 11:12 PM
newslang
[SOLVED] An Applied Trig/Max and Min Problem
Hi there,

I'm currently struggling with the following question from my textbook.

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

I know that I need to find the area and then the derivative and set it to zero but I'm not sure how to find the base and height of the triangle to get the area.

Thanks!
• Jan 10th 2012, 11:21 PM
MarkFL
Re: An Applied Trig/Max and Min Problem
Use the formula for the area A of a triangle:

$A=\frac{1}{2}ab\sin\theta$

You don't even need calculus to find the angle that maximizes A...
• Jan 10th 2012, 11:30 PM
newslang
Re: An Applied Trig/Max and Min Problem
Okay, right. I was just over thinking it.

So then, it's just $\pi/2$?

Thanks again MarkFL
• Jan 10th 2012, 11:32 PM
MarkFL
Re: An Applied Trig/Max and Min Problem
you got it! ;)
• Jan 10th 2012, 11:54 PM
CaptainBlack
Re: An Applied Trig/Max and Min Problem
Quote:

Originally Posted by MarkFL2
you got it! ;)

Except that cannot form a triangle with both A and B on the circle and andgle OBA=90 degrees!.

Draw a picture!

CB
• Jan 10th 2012, 11:56 PM
CaptainBlack
Re: An Applied Trig/Max and Min Problem
Quote:

Originally Posted by MarkFL2
Use the formula for the area A of a triangle:

$A=\frac{1}{2}ab\sin\theta$

You don't even need calculus to find the angle that maximizes A...

What are a and b, how are they related to the angle OBA?

Also do not use the same symbol for two different thing in the same problem: A - area of and label of a vertex of the triangle.

CB
• Jan 11th 2012, 12:05 AM
MarkFL
Re: An Applied Trig/Max and Min Problem
My apologies for my sloppiness.

Using coordinate geometry:

Putting point O at (0,0), point A at (5,0) and B at $\left(5\cos\theta,5\sin\theta\right)$ where $\theta=\angle AOB$ then the area T of the triangle is:

$T=\frac{1}{2}(OA)(OB)\sin(\theta)$
• Jan 11th 2012, 12:13 AM
CaptainBlack
Re: An Applied Trig/Max and Min Problem
Quote:

Originally Posted by MarkFL2
My apologies for my sloppiness.

Using coordinate geometry:

Putting point O at (0,0), point A at (5,0) and B at $\left(5\cos\theta,5\sin\theta\right)$ where $\theta=\angle AOB$ then the area T of the triangle is:

$T=\frac{1}{2}(OA)(OB)\sin(\theta)$

OK the area is maximised when your $\theta=\pi/2$ , but that is not what was asked for, what was asked for is angle OBA (which is now trivial to find, but your original presentation cannot have been anything but confusing to the OP).

CB