To determine [tex]\(\cot \theta\)[/tex] when [tex]\(P(x, y)\)[/tex] is a point on the unit circle at angle [tex]\(\theta\)[/tex], let’s follow the fundamental trigonometric definitions and the properties of the unit circle.
1. Understand the unit circle:
The unit circle is a circle with a radius of 1 centered at the origin (0,0) in the Cartesian coordinate system. Any point [tex]\(P(x, y)\)[/tex] on the unit circle lies at an angle [tex]\(\theta\)[/tex] from the positive [tex]\(x\)[/tex]-axis.
2. Trigonometric definitions on the unit circle:
- Sine ([tex]\(\sin \theta\)[/tex]) is defined as the [tex]\(y\)[/tex]-coordinate of point [tex]\(P\)[/tex].
- Cosine ([tex]\(\cos \theta\)[/tex]) is defined as the [tex]\(x\)[/tex]-coordinate of point [tex]\(P\)[/tex].
Thus, [tex]\( \sin \theta = y \)[/tex] and [tex]\( \cos \theta = x \)[/tex].
3. Tangent and cotangent:
- Tangent ([tex]\(\tan \theta\)[/tex]) is defined as the ratio of the sine and cosine. Therefore, in terms of [tex]\(x\)[/tex] and [tex]\(y\)[/tex],
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}
\][/tex]
- Cotangent ([tex]\(\cot \theta\)[/tex]) is the reciprocal of tangent:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} = \frac{x}{y}
\][/tex]
This is derived from the definitions above, substituting the appropriate values for sine and cosine.
Thus, based on the trigonometric relationships on the unit circle:
[tex]\[
\cot \theta = \frac{x}{y}
\][/tex]
Therefore, the correct option is [tex]\(B\)[/tex]:
[tex]\[
\boxed{\frac{x}{y}}
\][/tex]
