If $P(x, y)$ is the point on the unit circle determined by real number $\theta$, then $\tan \theta$ =
A. $\frac{1}{x}$
B. $\frac{1}{y}$
C. $\frac{y}{x}$
D. $\frac{x}{y}$
Answer (1)
Recognize that on the unit circle, x = cos θ and y = sin θ .
Recall the definition of tangent: tan θ = c o s θ s i n θ .
Substitute x and y into the tangent equation: tan θ = x y .
Conclude that tan θ = x y .
Explanation
Problem Analysis The problem states that P ( x , y ) is a point on the unit circle determined by a real number θ . We need to find an expression for tan θ in terms of x and y .
Coordinates on the Unit Circle Recall that on the unit circle, the coordinates of a point P corresponding to an angle θ are given by:
x = cos θ y = sin θ
Definition of Tangent The tangent function is defined as the ratio of the sine to the cosine:
tan θ = c o s θ s i n θ
Substitute and Find the Answer Substitute x and y into the expression for tan θ :
tan θ = x y
Final Answer Therefore, tan θ = x y .
Examples
Imagine you're tracking a friend running around a circular track with a radius of 1 unit. If you know your friend's coordinates (x, y) at any given moment, you can determine the angle θ they've swept from the starting point. The tangent of this angle, tan ( θ ) = x y , helps you understand the ratio of their vertical displacement (y) to their horizontal displacement (x) relative to the center of the track. This concept is useful in navigation, physics, and engineering to describe circular motion and angular relationships.
