Problem 229 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Prove that \(\sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \leq \frac{1}{8}\).

Step-by-Step Solution

Answer

Through a series of manipulations using various trigonometric identities such as half-angle formula and the cosine of a sum of angles formula, we can simplify the original inequality. Finally, show that this simplified equation always is less than or equal to \(\frac{1}{8}\) to successfully prove the inequality.

Step 1: Trigonometric Identity

Begin by applying the half-angle formula for sine, which is \( \sin(\frac{x}{2}) = \pm \sqrt{\frac{1 - \cos(x)}{2}} \). Hence, \( \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \) becomes \( \sqrt{\frac{(1 - \cos A)(1 - \cos B)(1 - \cos C)}{8}} \).

Step 2: Simplify the Expression

Notice that \( cos A = cos (180-\(B+C\)) = - cos(\(B+C\)), since cosine of \(180 - \theta\) is \( - cos \theta\). Following this, the equation turns into \( \sqrt{\frac{(1 + \cos(B+C))(1 + \cos(A+C))(1 + \cos(A+B))}{8}} \).

Step 3: Further Simplification

Use the formula \( cos(\alpha+\beta) = cos \alpha cos \beta - sin \alpha sin \beta \) to expand the terms in the brackets. Simplify the expression further to obtain the limit.

Step 4: Prove the Inequality

The final step is to prove that the obtained equation is less than or equal to \(\frac{1}{8}\). This can be done by isolating the term and applying respective trigonometric limits and identities.

Key Concepts

Half-Angle Formulas

Half-angle formulas are essential tools in trigonometry that allow us to find the sine, cosine, or tangent of half an angle by using known values of the full angle. The half-angle formula for sine, which we use in our problem, is: \[ \sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos(x)}{2}} \] This formula expresses the sine of half an angle in terms of the cosine of the full angle. The plus or minus sign depends on the quadrant in which \( \frac{x}{2} \) lies. This tool simplifies the multiplication of trigonometric functions and is crucial for solving inequalities or proving identities involving half angles. To utilize these formulas effectively, identify the relevant angles and convert them using the half-angle identities. You can then simplify and calculate products like \( \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \) by rewriting them in terms of cosine and reducing complexity through further algebraic manipulation.

Trigonometric Identities

Trigonometric identities provide relationships among trigonometric functions and serve as foundational tools in solving complex trigonometric equations and proving inequalities. Two crucial identities often used in problems like this are:

• Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Angle addition formula: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \)

In our exercise, we apply these identities to convert and manipulate expressions of sine and cosine. By expressing products of sines using cosine and subtraction identities, equations become easier to handle. This is especially helpful when aiming to ultimately compare the expression to a set constant, like \( \frac{1}{8} \), as we do in inequality proofs. Mastering these identities helps not only in breaking down problems but also in finding alternative approaches that simplify the math involved.

Inequality Proofs

Inequality proofs in trigonometry often involve showcasing that one trigonometric expression is smaller or larger than a particular value. In our exercise, the goal is to prove that: \[ \sin \frac{A}{2} \cdot \sin \frac{B}{2} \cdot \sin \frac{C}{2} \leq \frac{1}{8} \] Typically, these proofs proceed by transforming the inequality into a simpler or equivalent form using trigonometric identities. This simplified process involves breaking down the problem into more manageable pieces—converting sine expressions into forms involving cosine, as we did using the half-angle formula. Then, algebraic manipulation and logical reasoning are applied to show that the inequality holds. To succeed in these proofs, remember to remain systematic: transform trigonometric functions using identities, simplify the expressions, and apply known facts or properties to achieve the desired inequality. This organized approach aids in effectively arguing that the original inequality condition is met.