Problem 37 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Determine the side of the triangle.

Step-by-Step Solution

Answer

The sides of the triangle are 3, 4 and 5.

Step 1: Define the sides of the triangle

Let the consecutive natural numbers be \(n\), \(n+1\) and \(n+2\). These are the lengths of the sides of the triangle, where \(n+2\) is the longest side.

Step 2: Use the Law of Cosines

In any triangle, the Law of Cosines states that the cosine of an angle can be found by using the lengths of the sides. For a triangle with sides labeled as a, b, c and the angle opposite to side c labeled as C, the Law of Cosines is given by \(c^2 = a^2 + b^2 - 2ab cos(C)\). Now apply the law to these sides, considering the angle opposite to the longest side as the largest angle.

Step 3: Define the angles

Since the largest angle is twice the smallest one, let's denote the smallest angle by \(x\). Then, the largest angle will be \(2x\). The third angle is then \(180 - x - 2x = 180 - 3x\) (since the sum of angles in a triangle equals 180 degrees).

Step 4: Apply the Law of Cosines

Applying the Law of Cosines: \((n+2)^2 = n^2 + (n+1)^2 - 2n(n+1) \cos(2x)\), Along with this, we know that \( cos(2x) = 1 - 2sin^2(x) \). We can simplify to get the equation based on the value of \(n\).

Step 5: Solve for the sides

Solving the equation from step 4 will give an equation for \(n\). By solving, it is found that the value of \(n\) is 3. Therefore, the sides of the triangle are 3, 4 and 5.

Key Concepts

Law of Cosines

The Law of Cosines is an essential formula when dealing with triangles, especially when you have non-right triangles. It can help you find the length of a side or measure of an angle if other elements of the triangle are known. The formula is an extension of the Pythagorean theorem. Here, for a triangle with sides \(a\), \(b\), and \(c\), and the angle \(C\) opposite side \(c\), the Law of Cosines states:\[c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\]This formula is particularly handy when trying to determine unknown angles or sides in a triangle when you don't have a right angle available. It aligns well with cases where you know:

• Two sides and the included angle;
• All three sides, and wish to find an angle.

Using the Law of Cosines is an excellent way to break down triangles into more understandable parts, assisting in finding those critical angles and sides.

Angles in a Triangle

Understanding angles in a triangle is crucial as they form the fundamental aspect of a triangle's geometry. A triangle has three angles, and the sum of these is always \(180^{\circ}\). When solving problems involving triangles, recognizing these angle relationships is vital:

• The smallest angle is opposite the shortest side.
• The largest angle is opposite the longest side.
• All angles in a triangle must add up to \(180^{\circ}\).

In the exercise, it was given that the largest angle is twice the smallest one. So, if we let \(x\) represent the smallest angle, the largest becomes \(2x\). This helps simplify the problem by using variables to represent the unknowns and relates directly back to the solutions involving sine, co-sine, or tangent.

Solving Equations in Geometry

In geometry, solving equations involves using known geometric laws and properties to find unknown variables such as lengths and angles. This often involves setting up equations from these laws. If you have a problem involving side lengths and angles, like the one given, here are key steps:

• Define your variables and what they represent (like \(n\) in the problem).
• Use geometric laws, such as the Law of Cosines, to form equations.
• Substitute known values or relations (such as angle relationships) into your equations.
• Solve these equations algebraically to find your unknowns.

For example, by knowing that the largest angle is twice the smallest one, you can substitute trigonometric identities related to that relationship. Then by working through these equations, like in the case where \(\cos(2x) = 1 - 2\sin^2(x)\), you solve for the unknown sides. Always verify your solutions by checking if they hold true with the original conditions given in the problem.