Problem 17 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If the sides \(a, b, c\) of a triangle are in A.P., then find the value of \(\tan \left(\frac{A}{2}\right)+\tan \left(\frac{C}{2}\right)\) in terms of \(\cot \left(\frac{B}{2}\right)\)

Step-by-Step Solution

Answer

The value we are asked to find is \(2 \cot \left( \frac{B}{2} \right)\). After performing the necessary substitutions, we find that this simplifies to \(2 \cot \left( \cos^{-1}( \frac{a^2 + c^2 - b^2}{2ac} ) \right)\).

Step 1: Recall Properties of Triangles

The first step is to recall some basics about triangles whose sides form an arithmetic progression (A.P.). In such triangles, the sides \(a\), \(b\), and \(c\) would satisfy \( b = \frac{a + c}{2} \) . This is the characteristic property of an arithmetic sequence: all terms are equidistant from each other.

Step 2: Express the given terms using known Trigonometric Identities

The next step is to manipulate the terms given in the exercise using well-known trigonometric identities. These identities can be derived or remembered from studying trigonometry. Specifically, we'll use the identity \(\tan(x) = \frac{1}{\cot(x)}\). Substituting values from the problem into this identity gives us: \[ \tan \left( \frac{A}{2} \right) + \tan \left( \frac{C}{2} \right) = \frac{1}{\cot \left(\frac{B}{2} \right)} + \frac{1}{\cot \left(\frac{B}{2} \right)} \], which simplifies to \[ 2 \cot \left( \frac{B}{2} \right) \]

Step 3: Substitute for B using the First Property

The next step is to substitute values using the property which relates the angle \(B\) to the sides of the triangle. We know that this can be written as \(\sin(B) = \frac{b}{2R}\), where \(R\) is the circumradius. Now use the sine rule which states \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R\), to find the value of \(B\), which is \(\cos^{-1}(\frac{a^2 + c^2 - b^2}{2ac})\). Substituting this value into our equation gives us our final result.

Key Concepts

Arithmetic Progression in Triangles

Understanding the arithmetic progression (AP) in the context of triangles can be fascinating, as it reveals a unique relationship between the lengths of the sides. When the sides of a triangle, labeled as a, b, and c, are in AP, it suggests that the middle term b is the average of a and c.

This characteristic means you can express one side in terms of the others: \[ b = \frac{a + c}{2} \.\] This property can be particularly useful when trying to solve problems involving triangles in trigonometry, as it gives us a direct relationship to work with. Using AP in triangles allows us to simplify complex trigonometric expressions and establish connections between angles and sides that may not be immediately obvious.

For example, in the given problem, knowing that sides are in AP helps us move towards expressing the tangents of half-angles in terms of the cotangent of another half-angle, utilizing this basic property of an arithmetic sequence.

Trigonometric Identities

Trigonometric identities are fundamental tools in solving trigonometry problems. These identities are equations involving trigonometric functions that hold true for all values within their domains. Being familiar with trigonometric identities is crucial as they help simplify and solve equations, and can transform complex expressions into simpler forms.

One such identity that is widely used and known is the reciprocal identity: \[ \tan(x) = \frac{1}{\cot(x)} \.\] In our exercise context, when tasked with finding the value of \(\tan\left(\frac{A}{2}\right)+\tan \left(\frac{C}{2}\right)\) in terms of \(\cot\left(\frac{B}{2}\right)\), the reciprocal identity allows us to restate the tangent terms as cotangents, which can then lead to a more manageable equation to work with. This is a good showcase of how identities can significantly streamline the process of solving trigonometry problems by providing alternative forms of the expressions we are working with.

Sine Rule in Triangles

The sine rule, also known as the law of sines, is a powerful theorem in trigonometry that relates the lengths of the sides of a triangle to the sines of its angles. This rule is especially useful in non-right-angled triangles.

The sine rule is expressed as: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} = 2R \.\] where R is the radius of the circumcircle of the triangle. In the context of our problem, applying the sine rule helps us find the relationship between the side b and the angle B, connecting these to the circumradius of the triangle.

Knowing the sine rule is particularly important when angles are involved, as it can provide a pathway to solve for unknowns or establish additional relationships. For instance, in the exercise given, using the sine rule helps in solving for angle B and then using it to find the required value of the trigonometric expression involving half-angles. This exemplifies the sine rule's versatility in addressing a variety of problems in trigonometry involving triangles.