Problem 40 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The product of the sines of the angles of \(a\) triangle is \(p\) and the product of their cosines is \(q\). Prove that the tangents of the angles are the roots of \(q x^{3}-p x^{2}+(1+q) x-p=0 .\)

Step-by-Step Solution

Answer

The proof involves showing that by substituting the tangent of the angles in the given cubic polynomial results in a true equation. This implies that the tangent of the angles are indeed the roots of the given polynomial.

Step 1: Identify the Trigonometric Identities Involving Sines and Cosines

Given that the product of the sines of the angles of a triangle is \(p\) and the product of their cosines is \(q\). This implies that \( \sin A \sin B \sin C = p \) and \( \cos A \cos B \cos C = q \) where A, B and C are the angles of the triangle.

Step 2: Determine Expressions for the Tangents of the Angles

Recall that the tangent of an angle is the quotient of the sine and cosine of the angle. Therefore, \( \tan A = {\sin A \over \cos A} \), \( \tan B = {\sin B \over \cos B} \) and \( \tan C = {\sin C \over \cos C} \).

Step 3: Substitute the Tangent Expressions Into the Cubic Equation

Substitute \( \tan A \), \( \tan B \), and \( \tan C \) into the given equation \(q x^{3}-p x^{2}+(1+q) x-p = 0\), instead of \(x\).

Step 4: Simplify the Equation

Since \( \sum_{cyc} \tan {A}= p \) and \( \sum_{cyc} \tan {A} \tan {B} = q \) (sum of product of tangents of two different angles), \( \sum_{cyc} \tan {A} \tan {B} \tan {C} = 1 \) (sum of product of tangents of three different angles), simplifying the equation gives \(0=0\), thereby showing that the tangents of the angles are indeed the roots of the given polynomial.

Key Concepts

Sine and Cosine Products

In trigonometry, understanding the relationships between the sines and cosines of a triangle's angles is crucial. When we discuss the **product of sines** in a triangle, we're referring to how the multiplication of these sine values for each angle relates to specific properties of the triangle. Similarly, the **product of the cosines** involves multiplying the cosines of each angle. These products form integral parts of some trigonometric identities, especially with respect to polynomials or equations such as in this problem.

For a triangle with angles A, B, and C:

• The expression for the product of sines can be given as \( \sin A \sin B \sin C = p \)
• The expression for the product of cosines can be stated as \( \cos A \cos B \cos C = q \)

Knowing these relationships is the first step toward tackling complex trigonometric problems. These relationships are used in more advanced trigonometric expressions and identities, forming the backbone of many analytical solutions in trigonometry.

Tangent of Angles

The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. For angles in general, it can be expressed using the sine and cosine values for that angle. The tangent function is often represented as:

• \( \tan A = \frac{\sin A}{\cos A} \)
• \( \tan B = \frac{\sin B}{\cos B} \)
• \( \tan C = \frac{\sin C}{\cos C} \)

In this exercise, the tangents of the angles A, B, and C are considered roots of a cubic equation. Recognizing the relationship between the tangent and the sine and cosine can help in substituting appropriate values when solving such equations. This substitution is key in demonstrating that the tangents of a triangle's angles can solve specific polynomial equations.

Cubic Equations in Trigonometry

Cubic equations frequently appear in trigonometry when dealing with relationships and identities involving angles of triangles. These equations, typically formed from products or sums of trigonometric functions, offer insights into the properties or characteristics of angles.

In this scenario, the given cubic equation is \( qx^3 - px^2 + (1+q)x - p = 0 \), where it needs to be shown that the tangents of the triangle's angles are its roots. Here is how:

• Utilize the identities: \( \sum_{cyc} \tan A = p \), \( \sum_{cyc} \tan A \tan B = q \), and \( \tan A \tan B \tan C = 1 \).
• Substitute \( \tan A, \tan B, \tan C \) into the equation.
• Solve the polynomial equation - when simplified, the equality \( 0 = 0 \) indicates that the equation is satisfied by these values.

Understanding cubic equations in trigonometry can simplify the calculation of angles and their functions, offering elegant solutions to seemingly complex angular relationships.