Understanding the Trigonometric Functions for the 'Standard Angle' θ at Point (10,0)

When dealing with trigonometric functions, it's essential to grasp the definitions and properties of these functions at different angles. In this context, we will explore the values of the six circular functions when the terminal point of an arc is located at (10,0). This scenario departs slightly from the typical standard angle definition but offers valuable insights into the behavior of trigonometric functions.

Typical Standard Angle Definition

Traditionally, a 'standard angle' is defined as an angle in the unit circle with its vertex at the origin and one of its arms coinciding with the positive x-axis. The operations are then performed as the angle is rotated counterclockwise from the x-axis.

Terminal Point (10,0) and the Trigonometric Functions

Given the initial setup where the arc's terminal point is at (10,0), we can infer several key details:

1. The Angle θ 0: Since the terminal point (10,0) lies on the positive x-axis, the angle θ formed with the x-axis is 0 degrees or 0 radians. In this case,

sin θ 0. Consequently, cosec θ is undefined(division by zero). cos θ 1. Thus, sec θ 1. tan θ 0. Hence, cot θ is undefined(again division by zero).

These definitions are consistent with the standard trigonometric function values at θ 0.

Implications and Flexibility in Trigonometric Definitions

While the scenario seems straightforward, it is important to recognize that the definitions of trigonometric functions are not limited to the unit circle. The values can be generalized to real numbers using the concept of the unit circle in the Cartesian plane. Here's how these generalizations work:

1. Sine Function (sin θ): The value of sin θ is the y-coordinate of the point on the unit circle that corresponds to the angle θ. At θ 0, the point (10,0) lies on the positive x-axis, giving a y-coordinate of 0.

2. Cosine Function (cos θ): The cos θ is the x-coordinate of the point on the unit circle. At θ 0, the x-coordinate of (10,0) is 10. However, if we consider the unit circle with a radius of 1, the x-coordinate would be 1.

3. Tangent Function (tan θ): The tangent is the ratio of the sine to the cosine, namely tan θ sin θ / cos θ. At θ 0, both numerator and denominator are zero, making the tangent undefined.

4. Cosecant Function (cosec θ): The cosecant is the reciprocal of the sine. At θ 0, the sine is zero, which makes the cosecant undefined.

5. Secant Function (sec θ): The secant is the reciprocal of the cosine. At θ 0, the cosine is 1, making the secant equal to 1.

6. Cotangent Function (cot θ): The cotangent is the reciprocal of the tangent, namely cot θ cos θ / sin θ. At θ 0, the tangent is undefined, thus making the cotangent undefined.

Conclusion

While the scenario of a standard angle θ at point (10,0) may seem unconventional, it offers an excellent opportunity to explore the underlying definitions and properties of trigonometric functions. Understanding these functions in various contexts, including non-unit circle scenarios, is crucial for deeper mathematical comprehension.