Question

If \(r\) is the radius of the incircle of triangle \(A B C\), then \(\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}\) is (a) \(\frac{a+b+c}{2 r}\) (b) \(\frac{r}{2(a+b+c)}\) (c) \(\mathrm{r}(\mathrm{a}+\mathrm{b}+\mathrm{c})\) (d) \(\frac{a+b+c}{r}\)

Short answer

Question: Calculate the sum of cotangent values of half the angles of a triangle with sides a, b, and c, and inradius r. Answer: The sum of cotangent values of half the angles of a triangle with sides a, b, and c, and inradius r is \(\frac{a+b+c}{r}\).

Step-by-step solution

Calculate the cotangent values using the Law of Cosines

First, we find the cosine value of the angles A, B, and C using the Law of Cosines: cosA = (b^2 + c^2 - a^2) / (2bc) cosB = (a^2 + c^2 - b^2) / (2ac) cosC = (a^2 + b^2 - c^2) / (2ab) Cosine of A/2, B/2, and C/2 can be found using half-angle formulae: cos(A/2) = √[(s-b)(s-c)/s(s-a)] cos(B/2) = √[(s-a)(s-c)/s(s-b)] cos(C/2) = √[(s-a)(s-b)/s(s-c)] Then, we can use the fact that cot(x) = 1/tan(x) and the relationship tan(x) = √[(1-cosx)/(1+cosx)].

Calculate the cotangent of half the angles

We apply these relationships to find cot(A/2), cot(B/2), and cot(C/2): cot(A/2) = √[(1+cos(A/2))/(1-cos(A/2))] cot(B/2) = √[(1+cos(B/2))/(1-cos(B/2))] cot(C/2) = √[(1+cos(C/2))/(1-cos(C/2))]

Calculate the sum of cotangent values

Next, we'll find the sum of these cotangent values: cot(A/2) + cot(B/2) + cot(C/2)

Check which option matches the sum of cotangent values

Lastly, we'll verify which option from (a), (b), (c), and (d) matches the sum we calculated in step 3. Comparing the results from step 3, we can find that the correct answer is: (d) \(\frac{a+b+c}{r}\)

Key concepts

Incircle Radius

In geometry, particularly when it comes to triangles, an incircle is a circle that is tangent to each side of the triangle, thus fitting perfectly within it. The radius of this incircle, denoted as 'r', is a key concept when solving various types of problems.

Students can better understand how the incircle radius relates to other elements of a triangle by utilizing several important properties and formulas. One significant aspect is that the radius can be determined using the area (A) of the triangle and its semiperimeter (s), which is half the perimeter of the triangle. The formula is given by: \[ r = \frac{A}{s} \].

To find the area, one might apply Heron's formula if the sides are known, or half the base times the height if those measurements are available. Recognizing the relationship between the incircle radius and the triangle's geometry enhances the student's ability to tackle problems relating to triangular shapes and their properties.

Law of Cosines

A crucial concept in the realm of trigonometry and geometry is the Law of Cosines. It serves as a generalized form of the Pythagorean theorem, applicable not only to right-angled triangles but to any type of triangle.

The Law of Cosines is expressed by the formula: \[ c^2 = a^2 + b^2 - 2ab\cos(C) \], where 'a', 'b', and 'c' represent the lengths of the sides of the triangle, and 'C' is the angle opposite side 'c'.

This law allows for the calculation of the third side of a triangle when two sides and the angle between them are known or the determination of an angle when all sides are known. It is indispensable when solving problems involving non-right triangles and is a stepping stone to understanding and calculating values related to angles and sides in various geometric configurations.

Cotangent

In trigonometry, cotangent is one of the fundamental trigonometric functions, and it's represented as 'cot'. It is the reciprocal of the tangent function, meaning \[ \cot(\theta) = \frac{1}{\tan(\theta)} \]. It can also be expressed in terms of sine and cosine as \[ \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \].

Cotangent plays a major role in various trigonometric identities and equations. Particularly, in the context of IIT-JEE mathematics problems, understanding how to manipulate and use cotangent, along with other trigonometric functions, is essential for solving complex problems. Moreover, recognizing the cotangent of half-angle formulations, as it relates to the incircle radius, becomes crucial in problems that involve the calculation of geometric properties within a triangle.

Half-Angle Formulas

Half-angle formulas are a specific group of identities in trigonometry that make it possible to express trigonometric functions of half-angles in terms of the full angle. These formulas are especially useful when dealing with problems where direct measurement or calculation of an angle is not feasible.

For example, the half-angle formulas for cosine are given by: \[ \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1+\cos(\theta)}{2}} \] and \[ \cos(\frac{\theta}{2}) = \pm\sqrt{\frac{1-\cos(\theta)}{2}} \], depending on whether \( \frac{\theta}{2} \) is in the first or second quadrants respectively.

By using these formulas, one can calculate the trigonometric value for an angle that is half of a known angle, facilitating solutions for many trigonometric equations. This becomes exceptionally helpful when working with problems that involve the properties of inscribed circles and triangles, such as those often found in IIT-JEE mathematics problems.