Problem 116 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(0\cos x\) (B) \(\cos (\sin x)<\cos x\) (C) \(\cos (\sin x)>\sin (\cos x)\) (D) \(\cos (\sin x)<\sin (\cos x)\)

Step-by-Step Solution

Answer

Options (A) and (C) are correct; \(\cos(\sin x) > \cos x\) and \(\cos(\sin x) > \sin(\cos x)\).

Step 1: Evaluate the Given Range

The exercise tells us that \(0 < x < \frac{\pi}{2}\). In this range, for any \(x\), we have the values \(0 < \sin x < 1\) and \(0 < \cos x < 1\). Both sine and cosine are decreasing functions in this interval, meaning if their input decreases, their output decreases.

Step 2: Compare \(\cos(\sin x)\) and \(\cos x\)

Since \(\sin x < x\) in the interval \(0 < x < \frac{\pi}{2}\), and knowing that cosine is a decreasing function, it follows that \(\cos(\sin x) > \cos x\). This makes option (A) correct.

Step 3: Compare \(\cos(\sin x)\) and \(\sin(\cos x)\)

In this interval, \(\cos x\) is such that \(0 < \cos x < 1\) and \(\sin(\cos x)\) produces a value between \(\sin 0 = 0\) and \(\sin 1\). Since \(\cos(\sin x) > \cos x\) and \(\cos x > \sin(\cos x)\) as cosine decreases slower than sine increases up to \(x = 1\), then \(\cos(\sin x) > \sin(\cos x)\), confirming option (C) as correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are essential in various fields, including physics, engineering, and computer science.Key trigonometric functions:

• Sine (\(\sin\)): In a right triangle, it is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\(\cos\)): This function is the ratio of the adjacent side's length to the hypotenuse.
• Tangent (\(\tan\)): The tangent of an angle is the ratio of the sine to the cosine of that angle.

Understanding these functions helps in solving problems involving angles and distances. All trigonometric functions are periodic and have specific properties within different intervals.

Sine and Cosine Relationships

The sine and cosine functions have a special relationship due to their periodic nature and definitions. They are often analyzed and compared, especially within specific intervals such as \(0 < x < \frac{\pi}{2}\).In this interval:

• Both \(\sin x\) and \(\cos x\) lie between 0 and 1.
• Both functions are decreasing as their input values increase within this range.
• Usually, \(\sin x\) is less than \(x\)
• This relationship is crucial when comparing values like \(\cos(\sin x)\) and \(\cos x\)

Exploring these relationships allows us to derive inequalities and solve equations, as seen in the given problem where \(\cos(\sin x)\) is compared with other expressions.

JEE Main Mathematics

The JEE Main Mathematics syllabus covers a range of topics, including trigonometric inequalities, which is crucial for aspirants aiming to succeed in this demanding examination. This subject requires not only solving complex equations but also understanding and applying mathematical principles such as trigonometric functions.In trigonometric inequalities:

• Students need to be adept at handling functions like \(\sin\) and \(\cos\).
• Familiarity with their properties and transformations is essential.
• Being able to deduce relationships such as \(\cos(\sin x) > \cos x\) or \(\cos(\sin x) > \sin(\cos x)\) is often tested.

Mastering these concepts is vital for answering questions efficiently and accurately, thus improving one’s score in the examination. Practicing exercises similar to the problem provided enhances readiness for the range of questions presented in the JEE Main Mathematics section.