[Math] Trigonometric Functions—Arbitrary Angle Definition

Question

Both the angle $\theta$ and the shaded triangle share the same adjacent and hypotenuse $-3/5$ since $\cos = \text{adjacent/hypotenuse}$. But, since we know $\cos(135^\circ)$ is not equal to $\cos(45^\circ)$, how should I interpret the shaded triangle angle?

Answer 1

Both the angle $\theta$ and the shaded triangle share the same adjacent and hypotenuse $-3/5$

This uses the definition of $\cos$ for the entire range $\theta \in [0, 2\pi)$ based on the unit circle, where the sides are considered as signed segments and, indeed, $\cos(\theta) = -3/5$.

since $\cos = \text{adjacent/hypotenuse}$.

This, on the other hand, uses the geometric definition of $\cos\,$ for acute angles, where the sides of the right triangle are considered as positive magnitudes. In this case, $\cos(\pi-\theta)=3/5\,$, indeed.

Quoting from wikipedia's Trigonometric functions - Right-angled triangle definitions:

In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total $180^\circ$ ($\pi$ radians). Therefore, in a right-angled triangle, the two non-right angles total $90^\circ$ ($\pi / 2$ radians), so each of these angles must be in the range of $(0, \pi/2)$ as expressed in interval notation. The following definitions apply to angles in this $0 – \pi/2$ range. They can be extended to the full set of real arguments by using the unit circle, or $\;\dots$