Problem 10 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If the circumference of the \(\Delta A B C\) lies on its incircle, then prove that, \(\cos A+\cos B+\cos C=\sqrt{2}\)

Step-by-Step Solution

Answer

The formulated exercise seems to contain an error. For an equilateral triangle, the correct expression would be \(\cos A+\cos B+\cos C=\frac{3}{2}\), not \(\sqrt{2}\). Please, check the problem statement.

Step 1: Identification

Identify the type of triangle. As the circumference of the triangle lies on the incircle, this means that the triangle is an equilateral triangle.

Step 2: Calculate individual angles

Since it is an equilateral triangle, all of its internal angles are equal and are \(60^{\circ}\) each.

Step 3: Compute cosine values

Calculate the cosines of the angles. Since all angles are equal, they will all have the same cosine value. We have: \(\cos A = \cos B = \cos C = \cos 60^{\circ} = \frac{1}{2}\).

Step 4: Sum up cosine values

Add the cosines of all angles. We get: \(\cos A + \cos B + \cos C = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}\).

Step 5: Simplify the sum

However, the target is to obtain the sum as \(\sqrt{2}\), which is not identical with the calculated value.

Step 6: Correcting calculation

It seems the problem formulation might be incorrect as the sum of cosines is not reaching to \(\sqrt{2}\). Thus the original problem statement is probable faulty, since for an equilateral triangle the sum of cosines of its angles equals to \(\frac{3}{2}\), not \(\sqrt{2}\).

Key Concepts

Equilateral Triangle

An equilateral triangle is a fascinating geometric figure where all sides are of equal length, and all angles are equal too. The internal angles of an equilateral triangle are always 60 degrees each. This means in triangle notation: \(A = B = C = 60^{\circ}\).
This uniformity makes equilateral triangles a key study object in geometry, often used to explore properties of symmetry and equal partitioning.
Moreover, these triangles feature consistent properties that simplify complex calculations:

• All sides are equal, making it a perfect example of symmetry.
• The angles' equality simplifies many trigonometric calculations.
• It is especially prominent in studying relationships with circles, such as incircles and circumferences.

Cosine Function

The cosine function is an essential part of trigonometry, used to relate the angles of a triangle to the lengths of its sides. In an equilateral triangle, because all angles are equal, calculating the cosine is straightforward.
For an angle \(\theta = 60^{\circ}\), the cosine function returns the value \(\cos 60^{\circ} = \frac{1}{2}\). This simplification stems from the unit circle concept where the cosine of an angle represents the horizontal component of that angle.
Some vital points about cosine in such contexts include:

• The uniform angles in an equilateral triangle means their cosine values are identical.
• In trigonometry, cosine not only helps in computations but also provides insights about geometric shapes.
• The step-by-step problem uses the cosine function to show how predictable and consistent values are in equilateral triangles.

Incircle

The incircle of a triangle is the largest circle that fits inside the triangle and touches all three sides. In an equilateral triangle, determining the incircle becomes easier due to symmetry. This circle is centered at the centroid of the triangle, which is also the point of intersection of many of its symmetries.
In an equilateral triangle:

• The radius of the incircle can be calculated easily using the formula \(r = \frac{a \sqrt{3}}{6}\), where \(a\) is the side length of the triangle.
• The point where the incircle touches a side is equidistant from its endpoints, creating equal segments.
• This unique feature of touching all sides gives it significant applications in solving problems with a geometric scope.

When the triangle's circumference lies on the incircle, this adds a layer of complexity, often used in theoretical exercises to explore deeper geometric properties.