Problem 76 - Algebra for JEE ADVANCED

Question

If \(\alpha, \beta, y\) are the angles of a triangle and the system of equations \(\cos (\alpha-\beta) x+\cos (\beta-\eta) y+\cos (\gamma-\alpha) z=0\) \(\cos (\alpha+\beta) x+\cos (\beta+\eta) y+\cos (\gamma+\alpha) z=0\) \(\sin (\alpha+\beta) x+\sin (\beta+\gamma) y+\sin (\gamma+\alpha) z=0\) has non-trivial solutions, then triangle is necessarily a. cquilateral b. isosceles c. right angled d. acute angled

Step-by-Step Solution

Answer

The triangle is necessarily equilateral.

Step 1: Analyze the Properties of a Triangle

First, recall that in a triangle, the sum of its interior angles is 180 degrees, which means \(\alpha + \beta + \gamma = \pi\). This property allows us to express any angle in terms of the other two.

Step 2: Simplify Using Given Equations

Consider the system of equations:1. \(\cos(\alpha-\beta)x + \cos(\beta-\gamma)y + \cos(\gamma-\alpha)z = 0\)2. \(\cos(\alpha+\beta)x + \cos(\beta+\gamma)y + \cos(\gamma+\alpha)z = 0\)3. \(\sin(\alpha+\beta)x + \sin(\beta+\gamma)y + \sin(\gamma+\alpha)z = 0\)Using angle sum properties: \(\cos(\gamma - \alpha) = -\cos(\alpha + \beta)\) and similar expressions for other terms, the trigonometric identities provide symmetry for each combination of angles in these equations, allowing simplification of conditions where angles are evenly distributed, pointing to equal angle sides.

Step 3: Apply Conditions for Non-trivial Solutions

For there to be non-trivial solutions, the determinant of the coefficients of the equations must be zero. This condition tells us that the system of equations depends on a certain symmetry or proportionality among the angles, leading to possible specific angle relations.

Step 4: Identify Triangle Type

Given the symmetry and potential proportionality from the step above, if a triangle's angles allow the determinant of our system to be zero, then the triangle must prominently feature these equal angles, indicating it is an equilateral triangle where all angles \(\alpha = \beta = \gamma\).

Key Concepts

Angle Properties in Triangles

Understanding the properties of angles within a triangle is crucial for solving many trigonometric problems. One of the fundamental properties is that the sum of all interior angles in a triangle is always 180 degrees or \pi\ radians. This can be expressed as \( \alpha + \beta + \gamma = \pi \).

This property allows us to express any angle in terms of the other two, which is particularly useful when dealing with trigonometric equations. For instance, if you know two angles in a triangle, you can easily find the third. This feature becomes handy as it simplifies solving or proving congruence and similarity in triangles. Moreover, it assists in establishing conditions for more complex systems like those given in trigonometric problems involving multiple triangle angles.

Determinants in Systems of Equations

Determinants are mathematical tools used to indicate whether a system of linear equations has a unique solution. When dealing with a system of trigonometric equations, the determinant of coefficients from these equations can help identify critical conditions.

In our problem scenario, for the system to have non-trivial solutions, the determinant must be zero. This zero determinant implies that the equations are not independent, reflecting a symmetry or relationship among them. In the case of triangles, this often brings to light properties such as equilateral configurations, because the equal angle sides fulfill the condition of a zero determinant through inherent symmetrical relationships.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three sides and all three internal angles are equal. Each angle in an equilateral triangle measures 60 degrees, or \( \frac{\pi}{3} \) radians.

The properties of equilateral triangles are particularly useful in solving equations related to symmetry and proportionality. In our exercise, the condition for a non-trivial solution in the system of equations suggests that the angles need to be equal, as this symmetry causes the determinant to become zero.

Equilateral triangles demonstrate perfect balance and symmetry, making them a cornerstone in understanding broader geometric and trigonometric concepts. They provide a vivid example of symmetry in trigonometry and can be used to show how specific angle relations generate consistent results within a trigonometric framework.

Symmetry in Trigonometry

Symmetry is a key concept in trigonometry and helps in reducing complex equations to simpler forms. In the context of trigonometric equations involving angles of triangles, symmetry can refer to equal angles leading to equal trigonometric ratios, such as equal cosines or sines of angles, as seen in our system of equations.

By understanding symmetry, we can deduce relationships between angles and sides that might not be immediately apparent. Symmetric properties often imply that certain aspects of the triangle or equations balance perfectly, pointing to special triangle types like equilateral triangles.

Recognizing these symmetrical patterns helps in predicting and solving equations more efficiently, leveraging the natural balance that such properties provide. Symmetry in trigonometry is not only aesthetically pleasing but also mathematically profound, streamlining complex problem-solving processes.