Review question

# As $\theta$ varies, what's the largest this ratio of areas can be? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9770

## Suggestion

1. Let $a > 0$. Sketch the graph of

$y=\dfrac{a+x}{a-x} \quad \text{for} \quad -a < x < a.$

Are there any noticeable features of this function over this interval?

1. Let $0 < \theta < \dfrac{\pi}{2}$. In the diagram below is the half-disc given by $x^2 + y^2 \leq 1$ and $y \geq 0$.

The shaded region $A$ consists of those points with $-\cos \theta \leq x \leq \sin \theta$. The region $B$ is the remainder of the half-disc.

Find the area of $A$.

Can we divide the area $A$ into regions where we can easily calculate the areas?

1. Assuming only that $\sin^2\theta + \cos^2\theta = 1$, show that $\sin\theta\cos\theta \leq \dfrac{1}{2}$.

Would it help to consider $(\sin\theta-\cos\theta)^2$?

1. What is the largest $\dfrac{\text{area of } A}{\text{area of } B}$ can be, as $\theta$ varies?

How does our work for (i) help us here?