Trigonometric functions, like sine, cosine, and tangent, extend beyond right triangles when we place them on the coordinate plane. An angle, often denoted as $\theta$, is formed by a rotating ray originating from the origin. The point where the ray intersects the unit circle (a circle with a radius of 1 centered at the origin) gives us the coordinates $(x, y)$. These coordinates are directly related to the trigonometric functions: $\cos(\theta) = x$, $\sin(\theta) = y$, and $\tan(\theta) = \frac{y}{x}$. This allows us to define trig functions for any angle, not just angles within a right triangle. Understanding this connection unlocks a deeper understanding of periodic phenomena and wave behavior.
The unit circle provides a visual and intuitive way to understand the cyclical nature of trigonometric functions. As the angle $\theta$ increases, the coordinates $(x, y)$ trace out the unit circle, and the values of sine, cosine, and tangent repeat every $2\pi$ radians (or 360 degrees). This periodicity is fundamental to many applications of trigonometry in physics, engineering, and computer science. By using the coordinate plane, we can analyze the signs of trig functions in different quadrants and solve complex trigonometric equations.
Match the term with its definition:
| Term | Definition |
|---|---|
| 1. Unit Circle | A. The ratio of the opposite side to the adjacent side. |
| 2. Sine | B. A circle with a radius of 1 centered at the origin. |
| 3. Cosine | C. The ratio of the y-coordinate to the radius. |
| 4. Tangent | D. The ratio of the x-coordinate to the radius. |
| 5. Angle in Standard Position | E. An angle whose initial side is the positive x-axis. |
In the coordinate plane, the _________ of an angle $\theta$ is defined as the x-coordinate of the point where the terminal side of the angle intersects the _________ circle. The sine of the angle is defined as the _________. The tangent is the ratio of _________ to x.
Explain how understanding trigonometric functions on the coordinate plane helps in modeling periodic phenomena like sound waves or tides.
Please log in to post your answer.
Log InEarn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! 🚀