Understanding Trigonometric Functions for the Angle in Standard Position Determined by the Point (15, -8)

In this article, we will explore the six trigonometric functions of an angle in standard position, which is determined by the point (15, -8) in a coordinate system. We will go through the steps to find these functions, including identifying the radius and coordinates, and then calculating the sine, cosine, tangent, cosecant, secant, and cotangent. Additionally, we will explain the geometric interpretation of these functions.

Identifying the Coordinates and Calculating the Radius

To begin, we need to recognize the coordinates of the point (15, -8) and determine the radius, which is the distance from the origin to this point.

The coordinates are: x 15 y -8

The formula for the radius r is:

r sqrt{x^2 y^2} r sqrt{15^2 (-8)^2} r sqrt{225 64} r sqrt{289} r 17

Calculating the Six Trigonometric Functions

Once we have the radius, we can calculate the six trigonometric functions as follows:

Sine (sin) - The ratio of the opposite side to the hypotenuse: sin theta frac{y}{r} frac{-8}{17} Cosine (cos) - The ratio of the adjacent side to the hypotenuse: cos theta frac{x}{r} frac{15}{17} Tangent (tan) - The ratio of the opposite side to the adjacent side: tan theta frac{y}{x} frac{-8}{15} Cosecant (csc) - The reciprocal of sine, which is the ratio of the hypotenuse to the opposite side: csc theta frac{r}{y} frac{17}{-8} -frac{17}{8} Secant (sec) - The reciprocal of cosine, which is the ratio of the hypotenuse to the adjacent side: sec theta frac{r}{x} frac{17}{15} Cotangent (cot) - The reciprocal of tangent, which is the ratio of the adjacent side to the opposite side: cot theta frac{x}{y} frac{15}{-8} -frac{15}{8}

Summary of the Six Trigonometric Functions

Thus, the values of the trigonometric functions for the angle determined by the point (15, -8) are:

sin theta -frac{8}{17} cos theta frac{15}{17} tan theta -frac{8}{15} csc theta -frac{17}{8} sec theta frac{17}{15} cot theta -frac{15}{8}

Geometric Interpretation

Geometrically, these values can be understood in terms of a right triangle formed by the point (15, -8) and the origin. The point (15, -8) lies in the fourth quadrant. The reference triangle has a horizontal side of 15 units, a vertical side of -8 units, and a hypotenuse of 17 units (since 8-15-17 is a Pythagorean triple).

Figure 1: Trigonometric functions for the point (15, -8) in a coordinate plane.

The reference angle is the angle between the line segment from the point to the origin and the x-axis. Drawing a perpendicular from this point to the x-axis forms this reference triangle. The signs of the trigonometric functions are determined by the quadrant in which the terminal side of the angle lies.

In summary, the six trigonometric functions provide a measure of the relationship between the sides of a right triangle and the angle in standard position, which is crucial in many fields including physics, engineering, and architecture.