Understanding the Triangle Problem

The problem provides a specific equation relating the sides and angles of a triangle ABC and asks us to determine the relationship between the sines of its angles, specifically sin A, sin B, and sin C. We need to analyze the given equation and use standard triangle formulas and rules to find this relationship.

The relationship between sin A, sin B, sin C is often directly linked to the relationship between the sides a, b, c through the Sine Rule.

Analyzing the Given Equation

The given equation is:

\(2{\rm{a}}{\sin ^2}\left( {\frac{{\rm{C}}}{2}} \right) + 2{\rm{c}}{\sin ^2}\left( {\frac{{\rm{A}}}{2}} \right) = 2{\rm{a}} + 2{\rm{c}} - 3{\rm{b}}\)

Applying Trigonometric Identities and Sine Rule

To work with the terms involving \(\sin^2\left(\frac{A}{2}\right)\) and \(\sin^2\left(\frac{C}{2}\right)\), we use the half-angle formulas for a triangle:

• For angle A: \(\sin^2\left(\frac{A}{2}\right) = \frac{(s-b)(s-c)}{bc}\)
• For angle C: \(\sin^2\left(\frac{C}{2}\right) = \frac{(s-a)(s-b)}{ab}\)

where \(s\) is the semi-perimeter of the triangle, calculated as \(s = \frac{a+b+c}{2}\).

The Sine Rule states that in any triangle ABC, the ratio of a side to the sine of its opposite angle is constant:

\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\)

where R is the circumradius of the triangle. This implies that the sides a, b, c are proportional to sin A, sin B, sin C.

Step-by-Step Solution Derivation

Substituting Half-Angle Formulas into the Equation

Substitute the half-angle formulas into the given equation:

\(2a \left( \frac{(s-b)(s-c)}{ab} \right) + 2c \left( \frac{(s-a)(s-b)}{bc} \right) = 2a + 2c - 3b\)

Simplifying the Equation

Cancel out the common factors 'a' in the first term and 'c' in the second term:

\(2 \frac{(s-b)(s-c)}{b} + 2 \frac{(s-a)(s-b)}{b} = 2a + 2c - 3b\)

Notice that both terms on the left side have a common factor of \(\frac{2(s-b)}{b}\). Factor this out:

\(\frac{2(s-b)}{b} \left[ (s-c) + (s-a) \right] = 2a + 2c - 3b\)

Simplify the expression inside the brackets:

\(\frac{2(s-b)}{b} [2s - (a+c)] = 2a + 2c - 3b\)

Recall that \(2s = a+b+c\). Substitute this into the bracketed term:

\(\frac{2(s-b)}{b} [(a+b+c) - (a+c)] = 2a + 2c - 3b\)

\(\frac{2(s-b)}{b} [b] = 2a + 2c - 3b\)

Cancel out 'b' (assuming \(b \neq 0\), which is true for a triangle side):

\(2(s-b) = 2a + 2c - 3b\)

Now, substitute the full expression for \(s = \frac{a+b+c}{2}\):

\(2\left(\frac{a+b+c}{2} - b\right) = 2a + 2c - 3b\)

\(2\left(\frac{a+b+c-2b}{2}\right) = 2a + 2c - 3b\)

\(a+b+c-2b = 2a + 2c - 3b\)

\(a+c-b = 2a + 2c - 3b\)

Rearrange the terms to simplify the equation:

\(0 = (2a - a) + (2c - c) + (-3b + b)\)

\(0 = a + c - 2b\)

This gives us the relationship:

\(a + c = 2b\)

Interpreting the Result for Triangle Sides

The equation \(a + c = 2b\) is the defining condition for three numbers a, b, and c to be in an Arithmetic Progression (A.P.). Therefore, the sides of the triangle, a, b, and c, are in A.P.

Connecting Sides to Sines of Angles

From the Sine Rule, we know that \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k\), where \(k\) is a constant (\(2R\)). This implies that:

• \(a = k \sin A\)
• \(b = k \sin B\)
• \(c = k \sin C\)

Since the sides a, b, c are in A.P., they satisfy \(a+c=2b\). Substitute the expressions from the Sine Rule into this equation:

\(k \sin A + k \sin C = 2(k \sin B)\)

Assuming \(k \neq 0\) (which is true for any valid triangle), we can divide both sides by k:

\(\sin A + \sin C = 2 \sin B\)

This equation shows that \(\sin A, \sin B, \sin C\) satisfy the condition for being in an Arithmetic Progression (A.P.).

Conclusion on sin A, sin B, sin C Relationship

By simplifying the given equation using half-angle formulas and the definition of the semi-perimeter, we found that the sides a, b, c of the triangle are in A.P. Using the Sine Rule, which establishes a proportionality between the sides and the sines of the opposite angles, we concluded that sin A, sin B, sin C are also in A.P.

Revision Table: Key Concepts for Triangle Problems

Additional Information: Properties of Arithmetic Progression (A.P.)

An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant. This constant value is called the common difference. For example, 2, 4, 6 is an A.P. with a common difference of 2. The condition for three terms \(x, y, z\) to be in A.P. is that the middle term is the arithmetic mean of the first and third terms, i.e., \(y = \frac{x+z}{2}\), which rearranges to \(2y = x+z\). In our problem, we showed that the sides \(a, b, c\) satisfy \(a+c=2b\) and the sines of angles \(\sin A, \sin B, \sin C\) satisfy \(\sin A + \sin C = 2 \sin B\), confirming that both sets of values are in A.P.