The essence of trigonometric functions for acute angles lies in the fact that they are ratios of side lengths in a right triangle, defined by the angle size. The core logic is based onsimilar trianglesthe property: as long as an acute angle ∠A is fixed, the ratio of corresponding sides remains unchanged regardless of the size of the right triangle. This 'stability of ratios' enables the transition from 'geometric shapes' to 'algebraic values'.
Core Formula System
In $Rt\triangle ABC$, for a given acute angle $A$:
- Sine (Sine): $\sin A = \frac{\text{opposite side of } \angle A}{\text{hypotenuse}} = \frac{a}{c}$
- Cosine (Cosine): $\cos A = \frac{\text{adjacent side of } \angle A}{\text{hypotenuse}} = \frac{b}{c}$
- Tangent (Tangent): $\tan A = \frac{\text{opposite side of } \angle A}{\text{adjacent side of } \angle A} = \frac{a}{b}$
Example 2 Demonstration
In $Rt\triangle ABC$, $\angle C = 90^\circ$, $AB = 10$, $BC = 6$.
1. Identify the sides: opposite side $a = 6$, hypotenuse $c = 10$.
2. Use the Pythagorean theorem to find the adjacent side: $b = \sqrt{10^2 - 6^2} = 8$.
3. Calculate the ratios:
$\sin A = \frac{6}{10} = 0.6$;
$\cos A = \frac{8}{10} = 0.8$;
$\tan A = \frac{6}{8} = 0.75$.
🎯 Core Logic Summary
Definition: Regardless of the size of $Rt\triangle ABC$, as long as acute angle $A$ is given, the ratios of its sides are uniquely determined. When both A and B are acute angles, if $A \neq B$, then $\sin A \neq \sin B$, $\cos A \neq \cos B$, and $\tan A \neq \tan B$. This shows that function values correspond one-to-one with angle sizes.