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?

To derive the sine rule for angles $A$ and $C$ in triangle $ABC$, an altitude $h$ is dropped from vertex $B$ to side $b$. Which equation correctly expresses $h$ in two different ways using right-triangle trigonometry?

Fill in the Blank: A Ferris wheel completes one full rotation every 90 seconds. To model the height in degrees using $h(t)=A\cos (Bt)+D$ where $t$ is in seconds, the value of the parameter $B$ is ________.

Which of the following functions represents the graph of $y=\sin x$ after it has been translated 5 units up and then stretched vertically by a factor of 2?

An object's vertical position is modeled by the function $$y=10\sin \left(\frac{\pi }{4}(t-C)\right)+20$$ where $t$ is time in seconds. If the object is at its midline (principal axis) and moving upward at $t=1$, find the smallest non-negative value for the phase shift $C$.