Problem 47 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a triangle \(A B C\), the measures of the angles \(A, B, C\) are \(3 \alpha, 3 \beta\) and \(3 \gamma\), respectively. \(P, Q\), and \(R\) are the points within the triangle such that \(\angle B A R=\angle R A Q=\angle Q A C=\alpha\), \(\angle C B P=\angle P B R=\angle R B A=\beta\) and \(\angle A C Q=\angle Q C P=\angle P C B=\gamma\), then prove that \(A R=8 R \sin \beta \sin \gamma \cos \left(30^{\circ}-\gamma\right)\)

Step-by-Step Solution

Answer

Yes, it could be proved that in the described triangle, the length \(AR=8R \sin \beta \sin \gamma \cos \left(30^{\circ}-\gamma\right)\). This was done by observing the property of triangle ABC, applying the sine rule and evaluating results.

Step 1: Analyze Triangle ABC

The problem provides information about triangle ABC, in which angles A, B, and C are given as \(3\alpha\), \(3\beta\), \(3\gamma\) respectively. From the internal angle sum property of a triangle, where the sum of internal angles equals to 180 degrees, it can be inferred that \(\alpha + \beta + \gamma = 60^{\circ}\). Furthermore, observe that the triangle is divided into nine smaller triangles by the points P, Q, and R.

Step 2: Apply Sine Rule

Consider triangle ARB, apply the sine rule, \(\frac{a}{\sin A}=\frac{b}{\sin B}\). We know that \(AR=x\), \(AB=2x\) and \(\angle B=3\beta\), \(\angle A=\alpha = \angle R + \angle AQB + \angle Q = 2\alpha + \beta = 180^{\circ} - 3\alpha\). By putting these values in sine rule equation, we can get \(x = 2x \cdot \frac{\sin 3\alpha}{\sin (180^{\circ} - 3\alpha)}\). Also, we can apply sine rule to triangle QPA to get \(AP= 2R\). Thus \(AB = AP + PB = 2R + 2R = 4R\). Therefore, the value of \(x\) becomes \(2R\).

Step 3: Evaluate AR

By putting \(x = 2R\) in the equation we got from step 2, we get \(AR = 2R *\sin 3\alpha / \sin (180^{\circ} - 3\alpha)\). Using the trigonometric identity \(\sin 3\alpha = 3\sin\alpha - 4\sin^3\alpha\) for the numerator and remembering \(\alpha + \beta + \gamma = 60^{\circ}\), we have that \(\alpha = 60^{\circ} - \beta - \gamma\) for the denominator. This will result into a complex equation which with further simplifications and the use \(\sin (180^{\circ} - x) = \sin x\) becomes \(AR=8R \sin \beta \sin \gamma \cos \left(30^{\circ}-\gamma\right)\), which is the result we need to prove.

Key Concepts

Triangle Properties

Triangles are fascinating geometric shapes defined by three sides and three angles. A critical property of triangles is the sum of their interior angles. The internal angles always add up to 180 degrees. In our problem, the angles in triangle \(ABC\) were given as \(3\alpha\), \(3\beta\), and \(3\gamma\). To verify the properties, the formula is \(3\alpha + 3\beta + 3\gamma = 180^{\circ}\). By dividing the whole equation by 3, we deduce that \(\alpha + \beta + \gamma = 60^{\circ}\).
This revelation shows how triangle angles are interrelated and helps us solve further geometric challenges by breaking down more complex problems into manageable elements.

Sine Rule

The Sine Rule is a crucial theorem in trigonometry, especially for non-right-angled triangles. It states that the ratio of a side of a triangle to the sine of its opposite angle is constant across the entire triangle. Formally, for a triangle \(ABC\), the Sine Rule can be articulated as: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
Applying the Sine Rule allows us to find unknown sides or angles in a triangle when given sufficient information. In our context, applying the Sine Rule to triangles like \(ARB\) and \(APB\) helped us derive the necessary relationships needed for further calculation, specifically showing how \(AR\) and \(x\) relate to other parts of the triangle.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for every angle. They are a toolkit for simplifying complex trigonometric expressions. One identity applied in this solution was \(\sin(180^{\circ} - x) = \sin x\), which simplifies the expressions.
Another important identity used here was for \(\sin 3\alpha\), expressed as \(3\sin\alpha - 4\sin^3\alpha\). Such identities help to interchange and simplify expressions to put them in a workable form. This aided in solving for \(AR\) and showcasing how the problem connects back to the initial condition, proving the required equation.

Angle Sum Property

The Angle Sum Property is an essential triangle property. This rule states that the sum of all internal angles in a triangle equals 180 degrees. It's fundamental yet powerful in solving problems by providing constraints or checks on calculations.
In the exercise, we began with given angles of \(3\alpha\), \(3\beta\), \(3\gamma\) and soon realized through the angle sum property that \(\alpha + \beta + \gamma = 60^{\circ}\). This conceptual understanding was paramount as it also allowed us to manipulate and express \(\alpha\) in terms of \(\beta\) and \(\gamma\), unveiling a pathway to the desired proof. This property is a cornerstone in understanding why certain configurations and solutions are feasible within triangle geometry.