Question

If the sides \(a, b, c\) of a triangle \(A B C\), are in AP then \(\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}\) are in (a) AP (b) GP (c) HP (d) None of these

Short answer

Based on the given information and the solution steps provided, the tangent of half-angles for triangle \(ABC\) with sides in arithmetic progression is not in arithmetic progression (AP), geometric progression (GP), or harmonic progression (HP). The correct answer is (d) None of these.

Step-by-step solution

Find a relationship between the angles

Since \(a, b, c\) are in AP, we have \(b=a+d\) and \(c=a+2d\) for some constant \(d\). By the Law of Cosines, we can write the following equations involving the angles: \( a^2 = b^2 + c^2 - 2bc\cos{A}\), \( b^2 = a^2 + c^2 - 2ac\cos{B}\), \( c^2 = a^2 + b^2 - 2ab\cos{C}\).

Apply trigonometric formulas

We will use half-angle formulas for the tangent function: \(\tan{\frac{A}{2}} = \sqrt{\frac{1-\cos{A}}{1+\cos{A}}}\), \(\tan{\frac{B}{2}} = \sqrt{\frac{1-\cos{B}}{1+\cos{B}}}\), \(\tan{\frac{C}{2}} = \sqrt{\frac{1-\cos{C}}{1+\cos{C}}}\). Using the Law of Cosines equations from Step 1, we can rewrite the expressions for the tangent of half-angles: \(\tan{\frac{A}{2}} = \sqrt{\frac{a^2-(b^2 + c^2)}{2bc+a^2}}\), \(\tan{\frac{B}{2}} = \sqrt{\frac{b^2-(a^2 + c^2)}{2ac+b^2}}\), \(\tan{\frac{C}{2}} = \sqrt{\frac{c^2-(a^2 + b^2)}{2ab+c^2}}\).

Check if the expressions are in AP, GP, HP, or None

By observing the expressions for \(\tan{\frac{A}{2}},\tan{\frac{B}{2}},\tan{\frac{C}{2}}\) derived above, it is clear that they are not in any simple progression (AP, GP, or HP). Therefore, the correct answer is (d) None of these.

Key concepts

Arithmetic Progression (AP)

Understanding the concept of Arithmetic Progression (AP) is foundational to solving many algebraic problems involving sequences. An AP is essentially a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. This is notated as:

\forall n \textrm{ (where } n \textrm{ is the term number)}, \( a_n = a_1 + (n-1)d \)
Here, \( a_n \) represents the nth term of the sequence, \( a_1 \) is the first term, and \( d \) is the common difference. For example, the sequence 2, 4, 6, 8, ... is an AP where each term increases by 2. In the context of geometry, knowing this property can help us identify and solve problems related to the sides or angles of polygons that are related in such a straightforward manner.

Half-Angle Formulas

In trigonometry, half-angle formulas are a type of identities that allow us to express trigonometric functions of half angles in terms of the trigonometric functions of the full angle. For instance,

\( \tan(\frac{\theta}{2}) = \sqrt{\frac{1-\cos(\theta)}{1+\cos(\theta)}} \)
is the half-angle formula for tangent. These formulas are particularly useful in situations where we cannot directly measure or calculate the half angle, but we do have information about the full angle. It's crucial when dealing with problems involving the angles of a triangle where the relationships are not immediately apparent. This tool can simplify complex trigonometric expressions and is integral to solving a variety of geometry and calculus problems.

Law of Cosines

The Law of Cosines is a powerful principle in trigonometry, bridging the gap between angle and side measurements in a triangle. The law states:

\( c^2 = a^2 + b^2 - 2ab\cos(C) \)
where \( a \), \( b \), and \( c \) represent the sides of a triangle, and \( C \) is the angle opposite side \( c \). This relationship extends the Pythagorean theorem to non-right triangles and allows us to find an unknown side or angle when given two sides and the included angle or two angles and a side. It's particularly indispensable for problems where the involved angles and sides are not comprising a right angle, thus regular Pythagorean relations are not applicable. In the context of the given exercise, employing the Law of Cosines, we establish a relationship between the sides of the triangle and their opposing angles, a crucial step towards determining the types of progressions formed by half-angle tangents.