Definition of the Basic Trigonometric Functions
1. The Setup: Point on a Circle
We begin by building a geometric picture that will serve as the foundation for all trigonometric definitions.
- Place a point O at the origin of a coordinate plane.
- Draw an initial radius OA along the positive $x$-axis.
- Rotate OA by an angle $\alpha$ (counter-clockwise for positive angles) to obtain the terminal radius OB.
- Point B lies on a circle of radius $R$ centred at O, so $B$ has coordinates $(x;\, y)$.
2. Key Insight — The Ratios Do Not Depend on R
Why can we define trigonometric functions using any circle, not just the unit circle? Because the ratios $\dfrac{y}{R}$ and $\dfrac{x}{R}$ stay the same no matter how large or small $R$ is.
Proof by Similar Triangles
Take two circles of radii $R_1$ and $R_2$ centred at O. For the same angle $\alpha$, the terminal radius intersects the first circle at $B_1(x_1;\, y_1)$ and the second at $B_2(x_2;\, y_2)$.
Drop perpendiculars from $B_1$ and $B_2$ to the $x$-axis, forming right triangles $\triangle OB_1C_1$ and $\triangle OB_2C_2$.
These triangles share the angle $\alpha$ at O and both have a right angle at C, so they are similar (AA criterion). Therefore:
$$ \frac{y_1}{R_1} = \frac{y_2}{R_2} \qquad \text{and} \qquad \frac{x_1}{R_1} = \frac{x_2}{R_2} $$Conclusion: The ratios depend only on the angle $\alpha$, not on the radius $R$. This is what makes trigonometric functions well-defined.
Drag the slider to change the angle and observe how the two similar triangles change while the ratios remain equal.