Problem 49 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If the sides of a triangle are in A.P. and if its greatest angle exceeds the least angle by \(\alpha\), show that the sides areinthe ratio \((1-x): 1:(1+x)\), where \(x=\sqrt{\frac{1-\cos \alpha}{7-\cos \alpha}}\).

Step-by-Step Solution

Answer

The sides of the triangle in arithmetic progression with middle term as \(a + d\) and \(a + 2d\) are expressed in the ratio \(1 - x : 1 : 1 + x\) where \(x = \sqrt{\frac{1- \cos \alpha}{7- \cos \alpha}}\).

Step 1: Identify the sides of the triangle

Let the sides of triangle be a, a+d, a+2d where (a+d) is the middle term, according to the properties of an Arithmetic progression (A.P). a refers to the first term and 2d is the common difference.

Step 2: Apply Law of Cosines

Apply law of cosines in the triangle. According to the law of cosines, \(c^2 = a^2 + b^2 - 2ab \cos C\), where C is the angle opposite side c. Here, the greatest angle exceeds the smallest one by \(\alpha\), therefore we have two relations: \((a+2d)^2 = a^2 + (a+d)^2 - 2a(a+d) \cos \alpha\) and \(a^2 = (a+d)^2 + (a+2d)^2 - 2(a+d)(a+2d) \cos(\alpha + A)\). By simplifying both the equations, we get two relations \((a^2 + d^2 - 2ad \cos \alpha = 4d^2)\) and \((a^2 +(a+2d)^2 + a^2 - 2a(a+2d) \cos \alpha = d^2)\).

Step 3: Solve the equations

Solving the two equations obtained in step 2, we end up with another two expressions: \(\frac{d}{a} = \sqrt{\frac{1- \cos \alpha}{7- \cos \alpha}}\) and \(a+2d = a + a \sqrt{\frac{1- \cos \alpha}{7- \cos \alpha}}\), which can be further simplified to \(1- x : 1 : 1 + x\) where \(x = \sqrt{\frac{1- \cos \alpha}{7- \cos \alpha}}\).

Step 4: Substitute the ratio

Substitute \(x = \sqrt{\frac{1- \cos \alpha}{7- \cos \alpha}}\) in \(1 - x : 1 : 1 + x\) to express the ratio of sides of the triangle

Key Concepts

Arithmetic Progression in Geometry

In geometry, an arithmetic progression (A.P.) is a sequence of numbers where the difference between consecutive terms is constant. This sequence is recognizable as it follows the rule of being expressed as: \[ a, a+d, a+2d, \ldots \] where \(a\) is the first term and \(d\) is the common difference. In the context of a triangle, when we say the sides are in arithmetic progression, it means if you list the sides in increasing order, the difference \(d\) between any two successive sides stays the same.
This characteristic is crucial in geometrical problems because it can significantly simplify the analysis of the triangle's properties. Such progressions can help relate the sides to the angles, using the properties of angles and trigonometric identities to find ratios among the sides.
Understanding how arithmetic progression applies to geometry can also help in visualizing how changing one side proportionally affects the others, given the constant difference, making it easier to understand some more advanced geometric configurations.

Law of Cosines

The Law of Cosines is a fundamental principle in trigonometry, particularly useful in solving problems involving triangles when you know two sides and the included angle or all three sides. It extends the Pythagorean theorem to non-right triangles and states that for a triangle with sides \(a\), \(b\), and \(c\), and angle \(C\) opposite to side \(c\): \[ c^2 = a^2 + b^2 - 2ab \cos C \] This law helps calculate the length of one side of a triangle if you know the other two sides and the angle between them, which is crucial when you don't have a right angle.
In our exercise, we applied the Law of Cosines to connect the sides in arithmetic progression with angles, particularly to find expressions that either directly or indirectly involve \(\cos \alpha\). Since \(\alpha\) is the difference between the greatest and smallest angles, understanding and using the Law of Cosines reveals relationships between the side lengths and angles, ultimately leading us to calculate a specific side ratio.

Triangle Angle Relationships

Triangles have a unique property: the sum of their interior angles is always \(180^\circ\). This property aids in finding unknown angles when at least two angles are given. When examining the angle relationships, the difference between the greatest and least angle, denoted as \(\alpha\), can lead to insightful revelations about the triangle's configurations.- If two angles differ by \(\alpha\), this gives two equations that can help find specific angles using \(\alpha\). - With sides in arithmetic progression, identified as \((a, a+d, a+2d)\), the difference in angles helps in categorizing and solving for specific triangle configurations.
Understanding these relationships allows the application of different trigonometric rules, helping derive the solutions of some complex geometric setups. When sides of a triangle form an arithmetic progression, employing both the angle-sum property and angle differences helps one decipher the intricate side-angle relationships, eventually yielding the desired ratios among them.