Hello, reiward!
Each of two tangents from an external point to a circle is 3 cm long.
The smaller arc which they intercept is 112° 30' 26''.
Find the radius of the circle. Code:
* * *
* * A
* o
* r * * * 3
* *
* * θ * *
* C o - - - - * - - - - o P
* * * *
* *
* r * * * 3
* o
* * B
* * *
The center of the circle is $\displaystyle C$; its radius is $\displaystyle r.$
Tangents $\displaystyle PA = PB = 3$
$\displaystyle \angle ACB \,=\,112^o30'13'' \quad\Rightarrow\quad \theta \:=\:\angle ACP \:\approx\:56.2536^o$
In right triangle $\displaystyle CAP\!:\;\;\sin\theta \:=\:\frac{3}{r} \quad\Rightarrow\quad r \:=\:\frac{3}{\sin\theta}$
ThereforeL .$\displaystyle r \:=\:\frac{3}{\sin56.2536^o} \;\approx\;3.6\text{ cm}$