Problem 18 - A Complete Resource Book in Mathematics for JEE Main

Question

If \([\sin x]+[\sqrt{2} \cos x]=-3, x \in[0,2 \pi]([.]\) denotes the greatest integer function) then \(x\) belongs to (A) \(\left[\frac{5 \pi}{4}, 2 \pi\right]\) (B) \(\left(\frac{5 \pi}{4}, 2 \pi\right)\) (C) \(\left(\pi, \frac{5 \pi}{4}\right)\) (D) \(\left[\pi, \frac{5 \pi}{4}\right]\)

Step-by-Step Solution

Answer

\(x\) belongs to \((\pi, \frac{5\pi}{4})\).

Step 1: Understand the Greatest Integer Function

The greatest integer function, denoted by \[x\], outputs the largest integer less than or equal to \(x\). For instance, \[2.7\] = 2 and \[-1.3\] = -2. This function rounds down to the nearest integer. If \([\sin x] + [\sqrt{2} \cos x] = -3,\) we need to find the range of values for \(x\) that satisfy this equation.

Step 2: Analyze the Sine Function Range

Since \(\sin x\) ranges between -1 and 1, \([\sin x]\) will take the values 0, or -1 for different ranges of \(x\). Calculate where \([\sin x] = 0\) (when \(\sin x\) is non-negative) and where \([\sin x] = -1\) (when \(\sin x\) is negative).

Step 3: Analyze the Cosine Function Range and \(\sqrt{2} \cos x\)

The function \(\cos x\) ranges from -1 to 1, so \(\sqrt{2} \cos x\) will vary approximately from -1.414 to 1.414. Thus, \([\sqrt{2} \cos x] = 0, -1,\) or \(-2\) depending on the cosine value. Determine the range where these integer values occur by considering limits where \(\sqrt{2} \cos x\to 0\), \(-1\), or \(-2\).

Step 4: Combine Conditions

To satisfy \([\sin x] + [\sqrt{2} \cos x] = -3\), solve:\([\sin x] + [\sqrt{2} \cos x] = -3\) with possible combinations: \([-1] + [-2] = -3\). This means \(\sin x\) is negative and close to -1, and \(\sqrt{2} \cos x\) is also negative and close to -1.414.

Step 5: Determine x Ranges

The solution satisfies \(\sin x < 0\), so \(x\) is in \(\left(\pi, 2\pi\right)\). Given the conditions \(\sqrt{2} \cos x\approaches -2,\), which happens when \(\cos x ≈ -1\), evaluate within \(\left(\frac{3\pi}{2}, \frac{5\pi}{4}\right)\) where both sine and cosine conditions unite.

Step 6: Choose the Correct Interval

Compare the determined intervals with the choices given. Option \((\pi, \frac{5\pi}{4})\) corresponds to our calculation where both \(\sin x\) and \(\sqrt{2} \cos x\) simultaneously yield integer parts satisfying the equation.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in mathematics and have vast applications. These functions include sine (\( \sin \)) and cosine (\( \cos \)), which are periodic functions derived from the angles of a right triangle. Each of these functions has a range of values it can take:

• The sine function ranges from \(-1\) to \(1\), meaning that for any angle \(x\), \(\sin x\) will be somewhere in this range.
• Similarly, the cosine function also has a range of \(-1\) to \(1\)

These functions are used extensively in various fields, from engineering to physics, because they can model periodic phenomena like waves and oscillations. When working with trigonometric equations, it's crucial to understand these functions' properties, such as their periodicity and oscillation between their maximum and minimum values.

Sine and Cosine Analysis

To delve deeper into the sine and cosine analysis, it's important to realize how these functions influence our calculations. We're specifically interested in determining values of \(x\) when \([ \sin x ] + [ \sqrt{2} \cos x ] = -3\).
To solve this, we capitalize on the periodic nature of sine and cosine:

• The sine function is negative when \(x\) is from \(\pi\) to \(2\pi\)
• The cosine function, when multiplied by \(\sqrt{2}\), alters its range to approximately \(-1.414\) to \(1.414\)

We analyze these ranges further to determine specific intervals where each function contributes to satisfying the equation \([ \sin x ] = -1\) and \([ \sqrt{2} \cos x ] = -2\). This indicates regions where both functions output negative values approaching their minimums, guiding us to specific intervals for \(x\).

Mathematical Interval Notation

Mathematical interval notation is a concise way to write a range of numbers. It helps in defining the set of solutions for equations like the one we are tackling. By using parentheses or brackets, we can express open and closed intervals:

• Open intervals, like \((a, b)\), include all numbers between \(a\) and \(b\) but not the endpoints themselves.
• Closed intervals, such as \([a, b]\), include the endpoints along with all numbers in between.
• Semi-open intervals, like \((a, b]\) or \([a, b)\), include either \(a\) or \(b\) but not both.

In our original problem, determining the correct interval where our condition holds true requires understanding this notation to specify where the function outputs meet the equation's conditions. Recognizing and interpreting these notations accurately leads us to select the option \((\pi, \frac{5\pi}{4})\) for our solution.