Question

\text { If for a triangle } \mathrm{ABC},\left|\begin{array}{lll} \mathrm{a} & \mathrm{b} & \mathrm{c} \\ \mathrm{b} & \mathrm{c} & \mathrm{a} \\ \mathrm{c} & \mathrm{a} & \mathrm{b} \end{array}\right|=0, \text { then } \sin ^{3} \mathrm{~A}+\sin ^{3} \mathrm{~B}+\sin ^{3} \mathrm{C} \text { is }

Short answer

Question: Find the value of sin³A + sin³B + sin³C for a triangle ABC with the given determinant: $$\left|\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| = 0$$ Answer: \(\sin^3 A + \sin^3 B + \sin^3 C = 2\sin^2 B \sin C - 2\sin^2 A \sin B - 2\sin^2 A \sin C + \sin^3 B - \sin^3 A + \sin^3 C\)

Step-by-step solution

Solve the determinant

To begin, we need to find the value of the determinant, which is: $$\left|\begin{array}{ccc} a & b & c \\ b & c & a \\ c & a & b \end{array}\right| = a(b^2 + c^2 - a^2) - b(a^2 + c^2 - b^2) + c(a^2 + b^2 - c^2)$$

Simplify the determinant

Now we simplify the expression to get: $$ab^2 + ac^2 - a^3 - ab^2 - bc^2 + b^3 + ac^2 + ab^2 - c^3 = -a^3 + b^3 - c^3 + 2ab^2 + 2ac^2 - 2bc^2$$ Since the determinant is zero, we have: $$-a^3 + b^3 - c^3 + 2ab^2 + 2ac^2 - 2bc^2 = 0$$

Use sine rule for triangle

We know that for any triangle, the sine rule is given by: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Now, let's solve for a, b, and c and substitute into our determinant equation from the previous step: \(a = k \sin A\), \(b = k \sin B\), and \(c = k \sin C\) where k is a constant Substituting the values into our determinant equation, we get: $$-k^3 \sin^3 A + k^3 \sin^3 B - k^3 \sin^3 C + 2k^3 \sin^2 A \sin B + 2k^3 \sin^2 A \sin C - 2k^3 \sin^2 B \sin C = 0$$

Factor out the constant k

Factoring out the constant k, we get: $$k^3 (-\sin^3 A + \sin^3 B - \sin^3 C + 2\sin^2 A \sin B + 2\sin^2 A \sin C - 2\sin^2 B \sin C) = 0$$ Since the determinant is zero, the expression inside the parentheses must be zero as well (assuming k is non-zero): $$-\sin^3 A + \sin^3 B - \sin^3 C + 2\sin^2 A \sin B + 2\sin^2 A \sin C - 2\sin^2 B \sin C = 0$$

Rearrange the equation to find the sum of sin³ values

Rearrange the equation to find the sum of sin³ A, sin³ B, and sin³ C: $$\sin^3 A + \sin^3 B + \sin^3 C = 2\sin^2 B \sin C - 2\sin^2 A \sin B - 2\sin^2 A \sin C + \sin^3 B - \sin^3 A + \sin^3 C$$ Now the equation represents \(\sin^3 A + \sin^3 B + \sin^3 C\) for the given triangle ABC.

Key concepts

Determinants

Determinants are mathematical expressions used to analyze certain properties of matrices. In the context of triangles, we often use a determinant to check certain conditions between the sides. For the triangle ABC in our problem, we are given a determinant with side lengths as its entries. This determinant can be set to zero, implying certain relationships among the side lengths.

The determinant given is:

• \[\begin{vmatrix} a & b & c \b & c & a \c & a & b \\end{vmatrix}\]

The solution involves expanding this determinant to get a polynomial in terms of the side lengths \(a, b,\) and \(c\). Solving or simplifying this will help us further when we look to apply the trigonometry rules like the sine rule in later steps.

Sine Rule

The sine rule connects the sides of a triangle with the sines of its opposite angles, allowing us to express each side's length in terms of the sine of its respective angle. It holds for any triangle and is expressed as:

• \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

By using the sine rule, one can solve for the side lengths of a triangle when the angles are known. In our problem, by assuming \(a = k \sin A\), \(b = k \sin B\), and \(c = k \sin C\), where \(k\) is some constant, we facilitate substitution into the determinant equation from earlier. This substitution enables us to transform a problem involving sides into one involving angles, giving us a different perspective and simplification.

Triangle Properties

Triangles have several properties and laws that can be applied to solve mathematical problems involving them. The foundational properties include:

• The sum of the internal angles of any triangle is \(180^\circ\).
• Each side of a triangle is shorter than the sum of the other two sides (the triangle inequality).
• Also, for any angle \(A\), \(\sin A\) and \(\cos A\) can describe the side length relative to the hypotenuse and adjacent sides in a right triangle, which proves useful in non-right triangles using laws like the sine and cosine rules.

Understanding these properties allows us to apply appropriate rules and theorems, such as when we use a determinant to analyze relationships between angles and sides.

Mathematics Problem-Solving

Solving mathematics problems requires a structured approach and understanding of applicable concepts. When approaching a triangle problem, we:

• Identify known values and what is being asked.
• Apply relevant mathematical rules like the sine rule, cosine rule, and properties of determinants where necessary.
• Simplify expressions using algebraic methods.

Revisiting our solution steps, simplifying the determinant expression allows us to equate terms and solve for unknowns. By substituting known relations (like those from the sine rule), we can transform complex expressions into simpler forms to deduce solutions. Effective problem solving often involves breaking down problems into smaller, manageable parts and making logical connections—key skills in mathematics.