Finding Trigonometric Function Values Given a Point on the Terminal Side of an Angle
In this article, we will explore how to determine the values of the trigonometric functions for an angle theta;, given that a point (x, y) is on its terminal side. We will use the example where P(-8, 6) is on the terminal side of the angle theta;.
Definition and Basics
The trigonometric functions are defined in terms of a point (x, y) on the terminal side of an angle with respect to the origin. Specifically, for an angle theta; with a point (x, y) on its terminal side, the radius (or distance from the origin to the point) r can be calculated as follows:
r sqrt{x^2 y^2}
Once r is known, the trigonometric functions can be determined using the following definitions:
Sine: sintheta; y/r Cosine: costheta; x/r Tangent: tantheta; y/x Cosecant: csctheta; 1/sin theta; r/y Secant: sectheta; 1/cos theta; r/x Cotangent: cottheta; 1/tan theta; x/yCalculation Steps
Step 1: Calculate the Radius r
For the point P(-8, 6), we calculate the radius as follows:
r sqrt{(-8)^2 6^2} sqrt{64 36} sqrt{100} 10
Step 2: Calculate the Trigonometric Functions
Using the calculated radius, we can now determine the values of the trigonometric functions:
Sine: sintheta; 6/10 3/5 Cosine: costheta; -8/10 -4/5 Tangent: tantheta; 6/-8 -3/4 Cosecant: csctheta; 10/6 5/3 Secant: sectheta; 10/-8 -5/4 Cotangent: cottheta; -8/6 -4/3Summary of Trigonometric Functions
The values of the trigonometric functions for the angle theta; given the point P(-8, 6) on its terminal side are:
sintheta; 3/5 costheta; -4/5 tantheta; -3/4 csctheta; 5/3 sectheta; -5/4 cottheta; -4/3Additional Considerations
It is important to note that the trigonometric functions can be simplified using the Pythagorean theorem. In this specific case, the point P(-8, 6) forms a right triangle with the origin, where the hypotenuse is 10 (a known Pythagorean triple 6, 8, 10). This allows us to directly find the trigonometric values:
sintheta; 6/10 3/5 costheta; -8/10 -4/5 tantheta; 6/-8 -3/4 csctheta; 10/6 5/3 sectheta; 10/-8 -5/4 cottheta; -8/6 -4/3Visualization and Additional Insights
A helpful visualization is to draw a diagram of the point and the triangle formed with the origin. By plotting the point P(-8, 6), drawing a line to the origin, and then drawing horizontal and vertical lines to the axes, we can visually confirm the values of the trigonometric functions. This diagram not only aids in understanding the problem but also helps in solving related questions.
Conclusion
Understanding how to find the trigonometric functions of an angle given a point on its terminal side is crucial for solving many trigonometry problems. By using the definitions and the Pythagorean theorem, we can calculate the values of sine, cosine, tangent, and their reciprocals. Visualizing the problem through a diagram can further enhance the understanding and application of these concepts.