Problem 25 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The maximum value of \(\cos \alpha_{1} \cdot \cos \alpha_{2} \cdot \cos \alpha_{3} \ldots \ldots \ldots \ldots . . \cos \alpha_{n}\) under the restriction \(0 \leq \alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots \ldots \ldots \ldots \ldots, \alpha_{n} \leq \frac{\pi}{2}\) and \(\cot \alpha_{1} \cdot \cot \alpha_{2} \cdot \cot \alpha_{3} \ldots \ldots \ldots \ldots \ldots . \cot \alpha_{n}=1\), is (a) \(\frac{1}{2^{n / 2}}\) (b) \(\frac{1}{2^{n}}\) (c) \(\frac{1}{2^{n}}\) (d) 1

Step-by-Step Solution

Answer

(d) 1

Step 1: Write the restriction in an alternative form

We can rewrite \( \cot \alpha_n \) as \( \frac{1}{\tan \alpha_n} \) , so the condition becomes \( \frac{1}{\tan \alpha_1} \cdot \frac{1}{\tan \alpha_2} \cdot \frac{1}{\tan \alpha_3} \cdots \frac{1}{\tan \alpha_n} = 1\), which simplifies to \( \tan \alpha_1 \cdot \tan \alpha_2 \cdot \tan \alpha_3 \cdots \tan \alpha_n = 1 \).

Step 2: Connect the trigonometric function with the restriction

The product \( \cos \alpha_1 \cdot \cos \alpha_2 \cdot \cos \alpha_3 \cdots \cos \alpha_n \) has connection with \( \tan \alpha_n \), because \( \cos \alpha_n = \frac{\cos^2 \alpha_n}{\sin \alpha_n} \), so \( \cos \alpha_n = \frac{1}{\tan \alpha_n} \).

Step 3: Substitute and simplify

Substitute \( \frac{1}{\tan \alpha_n} \) with \( \cos \alpha_n \) into the restriction equation, we get \( \cos \alpha_1 \cdot \cos \alpha_2 \cdot \cos \alpha_3 \cdots \cos \alpha_n = 1 \). The maximum value of this product, according to the given conditions is \(1\)

Key Concepts

Trigonometric Functions

Trigonometric functions are vital tools in the world of mathematics, especially when dealing with angles in triangles and periodic phenomena. Their primary role is to relate the angles of a triangle to the lengths of its sides. Among these functions, we find sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), as well as their reciprocals: cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each of these functions has unique properties and relationships, and they can be used to solve a wide range of mathematical problems:

• The cosine function (\(\cos\)) measures the adjacent side to the hypotenuse ratio of a right-angled triangle.
• The tangent function (\(\tan\)) provides the ratio of the opposite to the adjacent sides of the triangle.
• Cotangent (\(\cot\)) is the reciprocal of the tangent function, defined as \(\cot \theta = \frac{1}{\tan \theta}\).

Trigonometric functions are more than theoretical concepts. They are employed in various practical fields such as physics, engineering, and even in advanced financial models. Recognizing their properties and interrelations is crucial for simplifying complex mathematical equations.

Cotangent

Cotangent, abbreviated as "cot," is a less commonly discussed trigonometric function but plays a significant role in advanced mathematical applications. It is defined as the reciprocal of the tangent function: \(\cot \theta = \frac{1}{\tan \theta}\).The cotangent function simplifies expressions involving triangles and periodic phenomena by converting division problems into multiplication problems, thanks to its reciprocal nature.### Key Properties of Cotangent The cotangent function has remarkable properties that are similar to other trigonometric functions:

• It is defined as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
• Its value is zero when \(\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots\)
• It is periodic with a period of \(\pi\), meaning its graph repeats every \(\pi\) units.

Understanding the cotangent function deepens the knowledge of trigonometry and aids in solving identity-based problems, which are often encountered in mathematical competitions and exams.

IIT JEE Mathematics

The IIT JEE, or Joint Entrance Examination, is an esteemed engineering entrance test conducted in India. It is known for its rigorous and challenging nature, particularly in the Mathematics section. This examination requires a profound understanding of mathematical concepts, including trigonometric identities and functions. The trigonometry section of IIT JEE Mathematics often tests students on their ability to manipulate and simplify expressions using core functions like sine, cosine, and tan, as well as their reciprocals, such as cotangent. Students must not only recall these identities but also apply them effectively to solve complex problems quickly:

• Questions may include finding the maximum or minimum values of trigonometric expressions.
• The ability to simplify trigonometric equations using identities is critical.
• Problem-solving speed and accuracy are essential due to time constraints during the exam.

Preparing for IIT JEE Mathematics requires extensive practice with solved examples and understanding each step of solutions to cultivate a solid foundation in advanced mathematical concepts. Knowing the deeper relations between trigonometric functions and their applications will greatly benefit students in achieving success in this competitive examination.