A student attempts to find an angle $A$ in a triangle with side lengths $a=20,b=10,$ and $c=8$. Their calculator returns a 'domain error'. What is the most likely mathematical reason for this?

A hiker travels $6$ km on a bearing of $040^{\circ }$ and then $10$ km on a bearing of $100^{\circ }$. What is the hiker's distance from the starting point to the nearest kilometer?

An explorer travels 10 km East (bearing $090^{\circ }$), then turns and travels 10 km on a bearing of $150^{\circ }$. How far is the explorer from her starting point?

If the cosine rule is written in the form $\cos A=\frac{b^2+c^2-a^2}{2bc}$, what is the resulting value of $\cos A$ if the triangle satisfies the condition $a^2=b^2+c^2$?

In triangle $DEF$, you are given $d=10$, $e=12$, and the included angle $\angle F=45^{\circ }$. The length of side $f$ to two decimal places is $\underline{}$.