Sine, Cosine and Tangent of Angles α and −α
Big Idea
When we rotate a point on the unit circle by angle α and by angle −α, the two resulting points are mirror images about the x-axis. This simple geometric fact gives us powerful identities that let us convert trigonometric functions of negative angles into functions of positive angles.
1. Geometric Setup on the Unit Circle
Consider the unit circle centered at the origin $O(0;\,0)$ with the initial point $P(1;\,0)$ on the positive x-axis.
- Point M₁ is obtained by rotating $P$ counter-clockwise by angle $\alpha$. Its coordinates are: $$M_1 = (\cos\alpha;\;\sin\alpha)$$
- Point M₂ is obtained by rotating $P$ clockwise by angle $\alpha$ (i.e., by angle $-\alpha$). Its coordinates are: $$M_2 = (\cos(-\alpha);\;\sin(-\alpha))$$
- Symmetry: The x-axis bisects angle $M_1OM_2$, so $M_1$ and $M_2$ are symmetric about the x-axis.
Key Observations from Symmetry:
- The x-coordinates (abscissas) of $M_1$ and $M_2$ are equal.
- The y-coordinates (ordinates) of $M_1$ and $M_2$ are opposite in sign.
Therefore: $M_1 = (\cos\alpha;\;\sin\alpha)$ and $M_2 = (\cos\alpha;\;-\sin\alpha)$
Interactive visualization — available in the full lesson
Drag the slider to change α and observe how M₁ and M₂ remain symmetric about the x-axis.