Twierdzenie sinusów (Sine Theorem)
The first page introduces the Twierdzenie sinusów (Sine Theorem) and its fundamental principles. This theorem is a cornerstone in trigonometry, especially useful for solving problems involving non-right triangles.
The sine theorem states that for any triangle ABC with sides a, b, and c, and opposite angles A, B, and C respectively, the following relationship holds:
a / sin A = b / sin B = c / sin C = 2R
Where R is the radius of the circumscribed circle of the triangle.
Definition: The Twierdzenie sinusów establishes a relationship between the sides of a triangle and the sines of their opposite angles.
Highlight: The constant ratio in the sine theorem is equal to the diameter (2R) of the triangle's circumscribed circle.
This page also presents the theorem in its alternative forms:
a = 2R sin A
b = 2R sin B
c = 2R sin C
Example: In a triangle ABC, if side a = 5 cm and angle A = 30°, we can find the radius R of the circumscribed circle using the formula: R = a / (2 sin A).
The sine theorem is particularly useful when solving triangles where we know a side and some angles, or when we need to find the radius of the circumscribed circle.