Question

In a triangle ABC, if \( r : R = 1 : 3 : 7 \), then \( \sin(A + C) + \sin B = \)

• A. \( 0 \)
• B. \( \sqrt{3} \)
• C. \( 1 \)
• D. \( 2 \)

Hint:

In geometry, certain standard trigonometric identities are used to relate the angles and sides of a triangle. Make sure to memorize these to simplify your calculations.

Answer 1

The Correct Option is D

Step 1: Understand the given ratio of the inradius and circumradius.
We are given that the ratio of the inradius \( r \) to the circumradius \( R \) is 1 : 3 : 7, i.e., \[ \frac{r}{R} = \frac{1}{3}. \] This ratio gives us insight into the relationship between the inradius and circumradius in a triangle. Step 2: Apply the trigonometric identity.
The given identity is: \[ \sin(A + C) + \sin B = 2. \] This identity is based on the geometric properties of the triangle, where \( A \), \( B \), and \( C \) represent the angles of the triangle, and the relationship between these angles and the radii of the triangle leads to the identity. Conclusion:
Thus, the correct value of \( \sin(A + C) + \sin B \) is 2.