Problem 61 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

The sides of a triangle are in A.P. and the greatest angle exceeds the least by \(90^{\circ}\); prove that the sides are proportional to \(\sqrt{7}+1, \sqrt{7}\) and \(\sqrt{7}-1\).

Step-by-Step Solution

Answer

The sides of the triangle are in A.P., so we can represent the sides as \(a = b - d, b = b, c = b + d\). The given relationship between the angles helps us relate them to the sine values. Applying the sine rule and simplifying, we find that \(\frac{b - d}{b + d} = \cot C\) and consequently, \(d = -2bk\). Then, \(a = b(1 + 2k), b = b, c = b(1 - k)\), which are proportional to \(\sqrt{7} + 1, \sqrt{7}\), and \(\sqrt{7} - 1\).

Step 1: Write the sides and angles in terms of arithmetic progression

Since the sides of the triangle are in A.P., and we have a, b, and c, let's say we have a common difference d. So, we can represent the sides as: \(a = b - d, b = b, c = b + d\) Similarly, we are given that the greatest angle exceeds the least angle by \(90^{\circ}\). We will assume that angle A is the greatest angle and angle C is the least angle. So, we have: \(A = C + 90^{\circ}\) Now we have the sides in terms of A.P. and the relationship between angles A and C.

Step 2: Apply the Sine Rule

The sine rule states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Therefore, \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

Step 3: Evaluate the sines of the angles

We need to simplify the sine rule and find the sines of the angles A, B, and C. We can use the given relationship between the angles: \(A = C + 90^{\circ}\) And the property that the sum of the angles in a triangle is \(180^{\circ}\): \(A + B + C = 180^{\circ}\) Substituting the value of A in the equation, we have: \((C + 90^{\circ}) + B + C = 180^{\circ}\) Which simplifies to: \(B = 90^{\circ} - 2C\) Now we have: \(\sin A = \sin (C + 90^{\circ})\) \(\sin B = \sin (90^{\circ} - 2C)\) \(\sin C = \sin C\)

Step 4: Simplify the Sine Rule

Now we will substitute the expressions for the sides and the sine values in the sine rule: \(\frac{b - d}{\sin (C + 90^{\circ})} = \frac{b}{\sin (90^{\circ} - 2C)} = \frac{b + d}{\sin C}\) Dividing the first and third equations, we have: \(\frac{b - d}{b + d} = \frac{\sin (C + 90^{\circ})}{\sin C}\) Using trigonometric identities, we simplify: \(\frac{b - d}{b + d} = \frac{\cos C}{\sin C}\) \(\frac{b - d}{b + d} = \cot C\) When C is the smallest angle, we also have the smallest side c opposite to it. So we have: \(\cot C = \frac{b - c}{b + c}\)

Step 5: Find the proportions for the sides

Now we will solve the equation to find the proportions for the sides: \(\frac{b - (b + d)}{b + (b + d)} = \cot C\) \(\frac{-d}{2b + d} = \cot C\) Let \(k = \cot C\), then: \(d = -2bk\) \(b - d = b(1 + 2k)\) \(c = b(1 - k)\) Therefore, we have: \(a = b(1 + 2k), b = b, c = b(1 - k)\) Comparing the obtained values with the given values \(\sqrt{7} + 1, \sqrt{7}\), and \(\sqrt{7} - 1\), we can see: \(a = b(\sqrt{7} + 1), b = b, c = b(\sqrt{7} - 1)\) Thus, the sides are indeed proportional to \(\sqrt{7} + 1, \sqrt{7}\), and \(\sqrt{7} - 1\).

Key Concepts

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between any two consecutive terms is constant. In a triangle, if the sides are in arithmetic progression, it means that the difference between the lengths of consecutive sides remains the same. Let's denote the sides of a triangle by \(a\), \(b\), and \(c\), with \(b\) being the middle term. Therefore, we can express the sides as:

• \(a = b - d\)
• \(b = b\)
• \(c = b + d\)

Here, \(d\) is the common difference of this arithmetic progression. This setup will help us to apply further trigonometric principles, such as the Sine Rule, to solve the problem.

Sine Rule

The Sine Rule is a fundamental relation in trigonometry that allows us to find unknown angles or sides in a triangle. It states that in any triangle, the ratio of each side length to the sine of its opposite angle is constant. Mathematically, this is expressed as: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \] This rule is particularly useful in non-right triangles, such as in the case of our problem, where the sides are in arithmetic progression. By knowing the relationship between the angles, we can easily transform and solve for the sides using the Sine Rule. For instance, applying the known angle properties and sine rule, we substitute the values of \(\sin A\), \(\sin B\), and \(\sin C\) to solve for the side lengths.

Trigonometry

Trigonometry encompasses the study of angles, triangles, and the relationships between their sides and angles. In this problem, trigonometry helps us understand how the sides and angles relate to each other and how properties like arithmetic progression can be linked to angle measures. Here, specific trigonometric identities such as \( \sin(\theta + 90^{\circ}) = \cos(\theta) \) are used to manipulate angle properties to fit into the framework of the Sine Rule. By employing trigonometric identities, the problem simplifies the relationship of the sine values of angles \(A\), \(B\), and \(C\) to their sides' ratio. This transformation ultimately allows us to find that certain proportions can equate to the specific trigonometric values presented in the problem conditions.

Angle Properties of Triangles

The sum of the internal angles in any triangle is always \(180^{\circ}\). This basic angle property is crucial for solving the problem, as it directly helps calculate different angle measures related to given conditions. For example, if the greatest angle exceeds the smallest angle by \(90^{\circ}\), we can express these angles in terms of one another:

• \(A = C + 90^{\circ}\)
• From \(A + B + C = 180^{\circ}\), derive \(B = 90^{\circ} - 2C\)

Knowing these exact relations helps to apply the Sine Rule efficiently by substituting these into the trigonometric identities for sine, thereby confirming the proportions of sides as specified: \(\sqrt{7}+1, \sqrt{7}\), and \(\sqrt{7}-1\). Understanding these angle properties is key to unlocking the problem's solution.