For the function $f\left(x\right)=\sin \left(x\right)$ on the unit circle, what is its minimum value?

• A. $1$
• B. $0$
• C. $-1$
• D. $\frac{1}{2}$

Answer: (C)

Verified step by step guidance

1. Recall that the function given is \(f(x) = \sin(x)\), which represents the sine of an angle \(x\) measured in radians on the unit circle.
2. Understand that the sine function corresponds to the y-coordinate of a point on the unit circle as the angle \(x\) varies from \$0\( to \(2\pi\) radians (or \)0$ to \(360^\circ\)).
3. Know the range of the sine function: since it represents the vertical coordinate on the unit circle, its values always lie between \(-1\) and \$1$, inclusive.
4. Identify the minimum value of \(\sin(x)\) by considering the lowest point on the unit circle's vertical axis, which occurs at the angle \(x = \frac{3\pi}{2}\) radians (or \(270^\circ\)), where the sine value is \(-1\).
5. Conclude that the minimum value of \(f(x) = \sin(x)\) on the unit circle is \(-1\).