The tangent of pi (tan π) is zero because at pi radians, the corresponding point on the unit circle lies on the negative x-axis, where the y-coordinate is zero.
Understanding Tangent and the Unit Circle
To understand why tan π = 0, let's revisit the definition of the tangent function in relation to 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.
-
An angle θ, measured counterclockwise from the positive x-axis, determines a point (x, y) on the unit circle.
-
The trigonometric functions sine (sin θ) and cosine (cos θ) are defined as follows:
- sin θ = y
- cos θ = x
-
The tangent function (tan θ) is defined as the ratio of sine to cosine:
tan θ = sin θ / cos θ = y / x
Evaluating tan π
Pi radians (π) corresponds to 180 degrees. Let's analyze the point on the unit circle at π radians:
-
Location on the Unit Circle: As stated in the reference, since pi lies on the negative x-axis, the coordinates of the point on the unit circle at π radians are (-1, 0).
-
x and y Coordinates: Therefore, x = -1 and y = 0.
-
Calculating tan π: Using the definition of the tangent function:
tan π = y / x = 0 / -1 = 0
Therefore, tan π = 0.
| Angle (Radians) | Angle (Degrees) | x-coordinate | y-coordinate | tan(angle) = y/x |
|---|---|---|---|---|
| 0 | 0 | 1 | 0 | 0/1 = 0 |
| π/2 | 90 | 0 | 1 | 1/0 = Undefined |
| π | 180 | -1 | 0 | 0/-1 = 0 |
| 3π/2 | 270 | 0 | -1 | -1/0 = Undefined |
| 2π | 360 | 1 | 0 | 0/1 = 0 |
