📚 Topic Summary
The general definition of trigonometric functions extends the familiar right-triangle definitions to angles of any size, positive or negative. Instead of relying on the sides of a right triangle, we use the unit circle. A point $(x, y)$ on the unit circle, corresponding to an angle $\theta$, allows us to define the trigonometric functions as follows: $\sin(\theta) = y$, $\cos(\theta) = x$, $\tan(\theta) = \frac{y}{x}$, $\csc(\theta) = \frac{1}{y}$, $\sec(\theta) = \frac{1}{x}$, and $\cot(\theta) = \frac{x}{y}$. These definitions apply to all angles, allowing for the analysis of periodic phenomena and wave behavior. Understanding these definitions is crucial for pre-calculus and beyond!
This framework avoids the limitations of acute angles in right triangles and unlocks a deeper understanding of trigonometric functions and their periodic nature.
🔤 Part A: Vocabulary
Match the term with its correct definition:
| Term |
Definition |
| 1. Unit Circle |
A. The ratio of the opposite side to the adjacent side. |
| 2. Sine ($\sin(\theta)$) |
B. A circle with a radius of 1 centered at the origin. |
| 3. Cosine ($\cos(\theta)$) |
C. The reciprocal of the sine function. |
| 4. Tangent ($\tan(\theta)$) |
D. The y-coordinate of a point on the unit circle. |
| 5. Cosecant ($\csc(\theta)$) |
E. The x-coordinate of a point on the unit circle. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
In the general definition of trigonometric functions, we use the ____1____. For any angle $\theta$, the ____2____ of $\theta$ is defined as the y-coordinate of the point where the terminal side of the angle intersects the unit circle. The ____3____ of $\theta$ is the x-coordinate of the same point. The tangent is the ratio of sine to ____4____. The reciprocal of sine is called ____5____.
🤔 Part C: Critical Thinking
Explain how the general definitions of trigonometric functions allow us to define these functions for angles greater than $90^{\circ}$. Why is this useful?
