If [tex]\( P(x, y) \)[/tex] is the point on the unit circle determined by real number [tex]\( \theta \)[/tex], then [tex]\( \tan \theta \)[/tex] is:

A. [tex]\( \frac{y}{x} \)[/tex]

B. [tex]\( \frac{x}{y} \)[/tex]

C. [tex]\( \frac{1}{x} \)[/tex]

D. [tex]\( \frac{1}{y} \)[/tex]



Answer :

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.