1. The problem involves understanding the trigonometric relationships in a right triangle with angle $\alpha$ at vertex A, opposite side $X$, adjacent side $Y$, and hypotenuse $R$. 2. The fundamental trigonometric functions for angle $\alpha$ are: - $\sin \alpha = \frac{X}{R}$ (opposite over hypotenuse) - $\cos \alpha = \frac{Y}{R}$ (adjacent over hypotenuse) - $\tan \alpha = \frac{X}{Y}$ (opposite over adjacent) 3. These ratios come from the definitions of sine, cosine, and tangent in a right triangle. 4. To find any side or angle, use these formulas and the Pythagorean theorem $R^2 = X^2 + Y^2$. 5. For example, if $\alpha$ and $R$ are known, then $X = R \sin \alpha$ and $Y = R \cos \alpha$. 6. This helps solve for unknown sides or angles in right triangles using trigonometry.