How to Determine the Trigonometric Function Values for an Angle on Standard Position with a Given Point

Trigonometry is a fundamental branch of mathematics that deals with the relationships between the sides and angles of triangles. When dealing with a specific point on the terminal side of an angle in standard position, we can calculate the values of the six trigonometric functions (sine, cosine, tangent, cotangent, cosecant, and secant). In this article, we will walk through the process of finding these values for the angle with a terminal side passing through the point ((-4, -3)).

Step-by-Step Guide

1. Determine the Radius r

The radius r is the distance from the origin to the given point. This can be calculated using the distance formula:

r √(x2 y2) √((-4)2 (-3)2)

Calculating the value:

r √(16 9) √25 5

2. Identify the Coordinates x and y

From the given point ((-4, -3)), we extract the x-coordinate as (-4) and the y-coordinate as (-3).

3. Calculate the Trigonometric Functions

The six trigonometric functions are found by applying their definitions:

Sine (sinθ): sinθ y/r (-3)/5 (-3/5) Cosine (cosθ): cosθ x/r (-4)/5 (-4/5) Tangent (tanθ): tanθ y/x (-3)/(-4) 3/4

4. Calculate the Reciprocal Functions

The reciprocal functions (cosecant, secant, and cotangent) are found by taking the reciprocals of the sine, cosine, and tangent functions, respectively:

Cosecant (cscθ): cscθ 1/sinθ 1/((-3/5)) (-5/3) Secant (secθ): secθ 1/cosθ 1/((-4/5)) (-5/4) Cotangent (cotθ): cotθ 1/tanθ 1/(3/4) 4/3

Summary of Values

The values of the six trigonometric functions for the angle whose terminal side passes through the point ((-4, -3)) are:

sinθ (-3/5) cosθ (-4/5) tanθ 3/4 cscθ (-5/3) secθ (-5/4) cotθ 4/3

Additional Insights

The point ((-4, -3)) lies on a circle with a radius of 5. Notably, the Pythagorean triple (3, 4, 5) helps confirm that the distances in this context adhere to the Pythagorean theorem x2 y2 r2, where x and y are the coordinates, and r is the radius.

x2 y2 (-4)2 (-3)2 16 9 25, which matches r2 52.

To visualize the ratios, write out each trigonometric function:

sinθ y/r (-3/5) cosθ x/r (-4/5) tanθ y/x 3/4 cscθ 1/sinθ (-5/3) secθ 1/cosθ (-5/4) cotθ 1/tanθ 4/3

This article provides a clear and concise guide to finding the trigonometric function values for a given point on the terminal side of an angle in standard position, using detailed explanations and clear calculations.