Final answer:
The statement about the ratio (a/sin A) being equal to the diameter of the circumscribed circle around triangle ABC is true, as according to the Law of Sines, the ratio of any side of a triangle to the sine of the angle opposite that side is consistent and equals the diameter of the triangle's circumcircle.
Step-by-step explanation:
The student's question pertains to a classical result in trigonometry known as the Law of Sines, specifically in relation to the circumscribed circle or circumcircle of a triangle. The gist of the Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is constant for all three sides and angles of the triangle. In the context of a circumscribed circle, this constant is equal to the diameter of the circumcircle.
To illustrate, let's suppose we have triangle ABC with sides a, b, and c, and angles A, B, and C. According to the Law of Sines, we have:
• a/sin(A) = 2R
• b/sin(B) = 2R
• c/sin(C) = 2R
Here, R is the radius of the circumcircle. Hence, the statement that the ratio (a/sin A) is equal to the diameter of the circumcircle is true, as the diameter is twice the radius (D=2R). Note that this relationship is used in a variety of geometric problems and proofs.
One clear example of this relationship is when dealing with isosceles triangles, where two sides are equal and thus their respective opposite angles are also equal, simplifying the application of the Law of Sines even further. This result is key when solving for unknown sides or angles in a triangle when given limited information.