UpStudy Free Solution:   

Step 1: Define the radius \(( r) \) For a point \(P( x, y) \) on the terminal side of an angle \(\theta \) in standard position, the radius \(r\) is given by the distance from the origin to the point \(P( x, y) \), which can be calculated using the Pythagorean theorem:

\[r = \sqrt { x^ 2 + y^ 2} \]

Step 2: Define the trigonometric functions Using the coordinates of the point \(P( x, y) \) and the radius \(r\), the trigonometric functions are defined as follows: - Sine of \(\theta \): \[\sin \theta = \frac { y} { r} \]

- Cosine of \(\theta \): \[\cos \theta = \frac { x} { r} \]

- Tangent of \(\theta \): \[\tan \theta = \frac { y} { x} \]

Additional Knowledge

1. Unit Circle: On the unit circle (where \(r = 1\)), the coordinates of point \(P( x, y) \) directly give the values of cosine and sine for angle \(\theta \). This simplifies the trigonometric functions to:

\(\sin \theta = y\), \(\quad \cos \theta = x\), \(\quad \text { and} \quad \tan \theta = \frac { y} { x} \)

2. Quadrants: The sign of the trigonometric functions changes depending on which quadrant the terminal side of the angle \(\theta \) lies in. This affects the values of sine, cosine, and tangent.

Knowing these relationships can enable you to solve a wide range of problems in geometry, physics, and engineering. To master these subjects and get help with your homework, try UpStudy Trigonometry Solver. With detailed, step-by-step explanations, UpStudy ensures you understand any topic and do your best in your studies. With UpStudy, the act of studying has been simplified!