Science:Math Exam Resources/Courses/MATH110/December 2012/Question 03 (d)/Solution 1

First we need to find a formula for $f(\theta )$, the area of the triangle. The area of a triangle is $(b)(h)/2$. See the labeled picture below:

The base of the triangle is a radius of the circle, which is equal to 1. We can find the height by a trig ratio. Using the blue right triangle, we can see that $\sin \theta =h/1$. So $h=\sin \theta$. Plugging these into our area formula, we get:

$\displaystyle f(\theta )=(1)(\sin \theta )/2=(\sin \theta )/2$

It is now easy to find the derivative,

$\displaystyle f'(\theta )=(\cos \theta )/2$

completing the question.