Funzione Secante e Cosecante
This page introduces the concepts of secant and cosecant functions in trigonometry, providing their definitions, formulas, and geometric interpretations.
The secante definizione is presented as the reciprocal of the cosine function. It is formally defined as the function that associates an angle α with the reciprocal of its cosine value, provided that the cosine of α is not zero.
Definition: sec(α) = 1 / cos(α), where cos(α) ≠ 0
Similarly, the cosecant function is defined as the reciprocal of the sine function. It associates an angle α with the reciprocal of its sine value, as long as the sine of α is not zero.
Definition: csc(α) = 1 / sin(α), where sin(α) ≠ 0
The page also provides a geometric interpretation of these functions using similar triangles. This approach helps visualize the relationships between secant, cosecant, and the more familiar sine and cosine functions.
Example: In a right-angled triangle OPH, where OH represents the hypotenuse:
- sec(α) corresponds to the ratio OH:OP
- csc(α) corresponds to the ratio OH:PH
The similarity of triangles OPH and OPS is used to derive the secante formula:
OH : OP = OP : OS
cos(α) : 1 = 1 : OS
Therefore, OS = 1 / cos(α) = sec(α)
Similarly, the similarity of triangles OHP and OPS' is used to derive the cosecante = 1/sen formula:
PH : OH = OH : OS'
sin(α) : 1 = 1 : OS'
Therefore, OS' = 1 / sin(α) = csc(α)
Highlight: The grafico secante and grafico cosecante can be visualized using these geometric relationships, providing a clear understanding of how these functions relate to the unit circle and right-angled triangles.
This comprehensive explanation of secant and cosecant functions forms part of the broader topic of relazioni fondamentali goniometria, which are crucial for understanding advanced trigonometric concepts and solving complex problems in mathematics and physics.
