Question

To verify that the values of trigonometric ratios of an angle do not vary with the lengths of the sides of the triangle.

Answer 1

Let's consider a triangle ABC with side vector $\overrightarrow{a}$ and $\overrightarrow{b}$Using dot product

$\overrightarrow{a}\cdot \overrightarrow{b}=|\overrightarrow{a}||\overrightarrow{b}|\cos \theta$

$\therefore \cos \theta =\frac{\overrightarrow{a}\cdot \overrightarrow{b}}{|\overrightarrow{a}||\overrightarrow{b}|}$

$=\hat{a}\cdot \hat{b}$

Clearly $\cos \theta$ is independent of $|\overrightarrow{a}|$, it only depends on unit vector along CB and CA.