Algebra 2 study guide: Defining trig functions using x, y, r

📚 Defining Trigonometric Functions Using x, y, and r

In trigonometry, we often use the unit circle to define trigonometric functions. However, we can generalize these definitions using any point $(x, y)$ on the Cartesian plane and its distance $r$ from the origin. This approach provides a more flexible way to understand trigonometric functions beyond the unit circle.

📜 History and Background

The use of coordinates to define trigonometric functions evolved from early geometric studies. Ancient mathematicians used chords of circles, but the coordinate system, developed later, allowed for a more algebraic and analytical approach. The connection between geometry and algebra, facilitated by coordinate geometry, led to the modern definitions of trigonometric functions.

📌 Key Principles

• 📐 Defining r: The distance $r$ from the origin to the point $(x, y)$ is given by the Pythagorean theorem: $r = \sqrt{x^2 + y^2}$.
• 📈 Sine Function: The sine of an angle $\theta$ is defined as the ratio of the y-coordinate to the distance r: $\sin(\theta) = \frac{y}{r}$.
• 📏 Cosine Function: The cosine of an angle $\theta$ is defined as the ratio of the x-coordinate to the distance r: $\cos(\theta) = \frac{x}{r}$.
• 🌱 Tangent Function: The tangent of an angle $\theta$ is defined as the ratio of the y-coordinate to the x-coordinate: $\tan(\theta) = \frac{y}{x}$.
• 🔄 Reciprocal Functions: We also have reciprocal trigonometric functions:
  • cosecant: $\csc(\theta) = \frac{r}{y}$
  • secant: $\sec(\theta) = \frac{r}{x}$
  • cotangent: $\cot(\theta) = \frac{x}{y}$

🌍 Real-world Examples

These definitions are crucial in various fields:

• 🧭 Navigation: Determining directions and distances.
• 💡 Engineering: Calculating angles and forces in structures.
• 🎮 Computer Graphics: Creating realistic images and animations.

✍️ Example Problems

Let's work through a few examples:

1. Example 1: Given the point $(3, 4)$, find all six trigonometric functions.
   • First, find $r$: $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
   • Then,
     • $\sin(\theta) = \frac{4}{5}$
     • $\cos(\theta) = \frac{3}{5}$
     • $\tan(\theta) = \frac{4}{3}$
     • $\csc(\theta) = \frac{5}{4}$
     • $\sec(\theta) = \frac{5}{3}$
     • $\cot(\theta) = \frac{3}{4}$
2. Example 2: Given the point $(-5, 12)$, find all six trigonometric functions.
   • First, find $r$: $r = \sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$.
   • Then,
     • $\sin(\theta) = \frac{12}{13}$
     • $\cos(\theta) = \frac{-5}{13}$
     • $\tan(\theta) = \frac{12}{-5} = -\frac{12}{5}$
     • $\csc(\theta) = \frac{13}{12}$
     • $\sec(\theta) = \frac{13}{-5} = -\frac{13}{5}$
     • $\cot(\theta) = \frac{-5}{12} = -\frac{5}{12}$

📝 Practice Quiz

🔑 Conclusion

Understanding trigonometric functions using x, y, and r provides a versatile and powerful tool for solving problems in mathematics, science, and engineering. By mastering these definitions, you can tackle a wide range of applications and gain a deeper understanding of trigonometry.