In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

Q.In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

Given: \(\sec(B + C) = 2\) So, \(\sec(B + C) = \sec 60^\circ\) \(\therefore B + C = 60^\circ \quad \text{(i)}\) Also, \(\sin(2B - C) = \frac{1}{2}\) So, \(\sin(2B - C) = \sin 30^\circ\) \(\therefore 2B - C = 30^\circ \quad \text{(ii)}\) Adding equations (i) and (ii): \[ B + C + 2B - C = 60^\circ + 30^\circ \Rightarrow 3B = 90^\circ \Rightarrow B = 30^\circ \] Substituting \(B = 30^\circ\) into equation (i): \[ 30^\circ + C = 60^\circ \Rightarrow C = 60^\circ - 30^\circ = 30^\circ \] Now, \[ A = 180^\circ - (B + C) = 180^\circ - (30^\circ + 30^\circ) = 180^\circ - 60^\circ = 120^\circ \] Therefore, the measures of angles A, B, and C are: A = 120°, B = 30°, C = 30°