Consider the following statements:

1. There exists no triangle ABC for which sin A + sin B = sin C.

2. If the angles of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio 1 : √3 : 2.

Which of the above statements is/are correct?

Concept:

$\frac{\mathrm{a}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{A}}=\frac{\mathrm{b}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{B}}=\frac{\mathrm{c}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{C}}=\mathrm{k}$

Calculations:

Consider, a, b, c are the sides of triangle and A, B, C are the angles of the triangle. 

We know that, $\frac{\mathrm{a}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{A}}=\frac{\mathrm{b}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{B}}=\frac{\mathrm{c}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{C}}=\mathrm{k}$

⇒$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{A}=\frac{\mathrm{a}}{\mathrm{k}},\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{B}=\frac{\mathrm{b}}{\mathrm{k}},\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{C}=\mathrm{c}\mathrm{k}$

Given , sin A + sin B = sin C

⇒ A + B = C.

⇒ a + b = c

This is not posible.  

Hence, there exists no triangle ABC for which sin A + sin B = sin C.

(2) Given, the angles of a triangle are in the ratio 1 : 2 : 3.

Consider the angles of a triangle are A = x, B = 2x and C = 3x.

We know that, Sum of the angles of the triangle is 180^°

Hence, x + 2x + 3x = 180

6x = 180°

x = 30º

Angles are A = 30º, B = 60º and C =  90º

We know that, $\frac{\mathrm{a}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{A}}=\frac{\mathrm{b}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{B}}=\frac{\mathrm{c}}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{C}}=\mathrm{k}$

  $\frac{\mathrm{a}}{\mathrm{s}\mathrm{i}\mathrm{n}30}=\frac{\mathrm{b}}{\mathrm{s}\mathrm{i}\mathrm{n}60}=\frac{\mathrm{c}}{\mathrm{s}\mathrm{i}\mathrm{n}90}=\mathrm{k}$

  $\frac{\mathrm{a}}{\frac{1}{2}}=\frac{\mathrm{b}}{\frac{\sqrt{3}}{2}}=\frac{\mathrm{c}}{1}=\mathrm{k}$

  $\frac{\mathrm{a}}{1}=\frac{\mathrm{b}}{\sqrt{3}}=\frac{\mathrm{c}}{2}=\mathrm{k}$ 

Hence, the angles of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio 1 : √3 : 2..