Equivalent Representations of Trigonometric Functions Practice Test

A spring’s position is modeled by $f(t)=10\sin!\left(2t+\frac{\pi}{3}\right)$ (centimeters), where $t$ is seconds. Letting $t=\theta$ gives the polar form $r=10\sin!\left(2\theta+\frac{\pi}{3}\right)$. On the coordinate plane, the sinusoid has amplitude $10$, period $\pi$, and its first maximum occurs at $t=\frac{\pi}{12}$. This periodic motion matches simple harmonic oscillation. Based on the description, how does the phase shift change when converting the function to polar form?

A vibration sensor reads $f(x)=3\sin!\left(\frac{x}{2}-\frac{\pi}{4}\right)$ (volts), where $x$ is time in seconds. Converting by setting $x=\theta$ gives $r=3\sin!\left(\frac{\theta}{2}-\frac{\pi}{4}\right)$. On the coordinate plane, the sinusoid has amplitude $3$, period $4\pi$, and crosses the midline at $x=\frac{\pi}{2}$. This periodic signal can represent a steady machine hum. Based on the description, what is the amplitude of the function given in polar form?