Trig invar — Trig ratios depend only on the angle · Principia

Depends on: AA similarity.

Statement

Let $\triangle ABC$ and $\triangle A'B'C'$ both be right triangles with $\angle C = \angle C' = 90^\circ$ , and suppose the corresponding acute angles are equal:

\angle A = \angle A' = \theta.

Following the "opposite / adjacent / hypotenuse" naming convention, write $a = BC$ (opposite to $\angle A$ ), $b = CA$ (adjacent to $\angle A$ , not the hypotenuse), $c = AB$ (hypotenuse), and analogously for $a'$ , $b'$ , $c'$ . Then the three ratios

\frac{a}{c} = \frac{a'}{c'},\qquad \frac{b}{c} = \frac{b'}{c'},\qquad \frac{a}{b} = \frac{a'}{b'}

depend only on the acute angle $\theta$ , not on the size of the triangle.

$Similarity-invariance of trig ratios: two right triangles of different sizes; provided the acute angle \theta is the same, the three ratios a/c, b/c, a/b are identical.$

Trig ratios depend only on the angle

Statement

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