Defining sine, cosine, and tangent using x, y, r in Algebra 2

📚 Understanding Sine, Cosine, and Tangent with x, y, and r

In trigonometry, especially when dealing with the unit circle and beyond, we can define sine, cosine, and tangent in terms of the coordinates of a point on a circle and the circle's radius. This is a powerful way to extend these trigonometric functions beyond the acute angles found in right triangles.

📜 A Brief History and Background

The concepts of sine, cosine, and tangent have ancient roots, tracing back to the study of chords in circles by Greek mathematicians like Hipparchus. However, the generalization to coordinate geometry and using x, y, and r to define these functions became more formalized with the development of analytic geometry by mathematicians like René Descartes.

🔑 Key Principles

Let's consider a circle centered at the origin (0, 0) with radius $r$. If $(x, y)$ is a point on the circle, and $\theta$ is the angle formed by the positive x-axis and the line segment connecting the origin to the point $(x, y)$, then we can define the trigonometric functions as follows:

• 📐 Sine ($\sin \theta$): The sine of the angle $\theta$ is defined as the ratio of the y-coordinate to the radius $r$. Mathematically, $\sin \theta = \frac{y}{r}$. It represents the vertical component of the point on the circle relative to the radius.
• 📏 Cosine ($\cos \theta$): The cosine of the angle $\theta$ is defined as the ratio of the x-coordinate to the radius $r$. Mathematically, $\cos \theta = \frac{x}{r}$. It represents the horizontal component of the point on the circle relative to the radius.
• ➗ Tangent ($\tan \theta$): The tangent of the angle $\theta$ is defined as the ratio of the y-coordinate to the x-coordinate. Mathematically, $\tan \theta = \frac{y}{x}$. It can also be expressed as $\tan \theta = \frac{\sin \theta}{\cos \theta}$. Note that the tangent is undefined when $x = 0$.

🌍 Real-World Examples

Let's look at some practical examples:

1. Example 1: Unit Circle
   

   Consider the unit circle, where $r = 1$. If a point on the unit circle is $(\frac{\sqrt{3}}{2}, \frac{1}{2})$, then $\theta = 30^{\circ}$ (or $\frac{\pi}{6}$ radians). Therefore:

   • $\sin 30^{\circ} = \frac{1/2}{1} = \frac{1}{2}$
   • $\cos 30^{\circ} = \frac{\sqrt{3}/2}{1} = \frac{\sqrt{3}}{2}$
   • $\tan 30^{\circ} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$
2. Example 2: Circle with Radius 5
   

   Consider a circle with radius $r = 5$. If a point on the circle is $(3, 4)$, then we can find the trigonometric functions as follows:

   • $\sin \theta = \frac{4}{5}$
   • $\cos \theta = \frac{3}{5}$
   • $\tan \theta = \frac{4}{3}$

📝 Practice Quiz

Answers: 1) 12/13, 2) -3/5, 3) 1

💡 Conclusion

Defining sine, cosine, and tangent using x, y, and r provides a versatile way to understand these trigonometric functions in various contexts, extending beyond the limitations of right triangles. This approach is fundamental in advanced mathematics and physics, enabling us to analyze periodic phenomena and oscillations.