How to do this without using trigonometry please?
A sheep is tethered to a post which is 6 m from a long fence. The length of rope is 9 m. Find the area which the sheep can feed on.
Let the post be $(0,0)$. Define such area to be $A_f$ then $A_f = A_c - A_s + A_t$ where:
$A_c$ is the area of the circle centered at $(0,0)$ with radius $r=9$
$A_s$ is the area of the sector with arc with endpoints $(-x,6)$ and $(x,6)$ having angle $\theta$
$A_t$ is the area of the triangle with vertices $(0,0)$, $(-x,6)$ and $(x,6)$
where $x$ is the positive number on the circle whose $y$-coordinate is $6$.
I got $x$, $A_c$ and $A_t$. How do we get $A_s$?
What I would do is
$A_s = \frac12 (\theta) (81m^2)$ where $\theta$ is obtained by
$\cos ( \frac{\theta}{2} ) = \frac69$
This is from IB Standard Level. Students know degree radian conversion, arc length $l = r \theta$, area of sector $\frac12 (\theta) (r^2)$, quadratics, functions, exponentials, logarithms, functions, sequences and series.
Students don't yet know $\sin$, $\cos$, trig stuff.