To determine the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(P(x, y)\)[/tex] is a point on the unit circle corresponding to the angle [tex]\(\theta\)[/tex], we need to understand the relationship between the coordinates of the point and the trigonometric functions.
1. Definition of Tangent:
The tangent of an angle [tex]\(\theta\)[/tex] in the context of the unit circle can be defined using the coordinates of the point [tex]\(P(x, y)\)[/tex].
2. Unit Circle Coordinates:
- The point [tex]\(P(x, y)\)[/tex] on the unit circle corresponds to an angle [tex]\(\theta\)[/tex] measured from the positive x-axis.
- By definition, the x-coordinate of point [tex]\(P(x, y)\)[/tex] on the unit circle is [tex]\(\cos \theta\)[/tex],
- and the y-coordinate is [tex]\(\sin \theta\)[/tex].
3. Formula for Tangent:
The tangent of an angle [tex]\(\theta\)[/tex] is defined as the ratio of the sine of the angle to the cosine of the angle:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
4. Applying Coordinates:
- Since [tex]\(P(x, y)\)[/tex] is on the unit circle, and [tex]\(x = \cos \theta\)[/tex] and [tex]\(y = \sin \theta\)[/tex],
- we can substitute these values into the formula:
[tex]\[
\tan \theta = \frac{y}{x}
\][/tex]
Thus, the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(P(x, y)\)[/tex] is on the unit circle is [tex]\(\frac{y}{x}\)[/tex].
So the correct answer is:
[tex]\[
\boxed{\frac{y}{x}}
\][/tex]
Therefore, [tex]\(\tan \theta = \frac{y}{x}\)[/tex], which corresponds to option A.
