sin/cos/tan — Definition of sin / cos / tan · Principia

Depends on: Trig ratios depend only on the angle, Pythagorean theorem.

Statement

Let $\triangle ABC$ be a right triangle with $\angle C = 90^\circ$ , and let $\theta = \angle A \in (0^\circ,\,90^\circ)$ be one of its acute angles. Following the "opposite / adjacent / hypotenuse" naming convention, write

a = BC \;(\text{opposite to } \theta),\qquad b = CA \;(\text{adjacent to } \theta,\text{ not the hypotenuse}),\qquad c = AB \;(\text{hypotenuse}).

The three trigonometric ratios of the acute angle $\theta$ are then defined as the pairwise quotients of these three sides:

\sin\theta \;:=\; \frac{a}{c} \;=\; \frac{\text{opp}}{\text{hyp}},\qquad \cos\theta \;:=\; \frac{b}{c} \;=\; \frac{\text{adj}}{\text{hyp}},\qquad \tan\theta \;:=\; \frac{a}{b} \;=\; \frac{\text{opp}}{\text{adj}}.

Well-definedness (that is, the fact that the three ratios above depend only on $\theta$ and not on the chosen right triangle) is delivered directly by Trig ratios depend only on the angle.

$Definition of sin/cos/tan: a right triangle \triangle ABC with acute angle \theta, sides a (opposite), b (adjacent), c (hypotenuse); the three ratios define \sin\theta=a/c, \cos\theta=b/c, \tan\theta=a/b.$

Definition of sin / cos / tan

Statement

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