Problem 12 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)\), then \(\frac{\sin (\alpha+\beta+\gamma)}{\sin \alpha+\sin \beta+\sin \gamma}\) is \((\mathrm{A})<1\) (B) \(>1\) \((\mathrm{C})=1\) (D) none of these

Step-by-Step Solution

Answer

The answer is (A) <1.

Step 1: Understanding the Range and Sine Function

We know that \( \alpha, \beta, \gamma \) are all angles in the first quadrant since they are between 0 and \( \frac{\pi}{2} \). In this range, the sine function is increasing, meaning as the angle increases, the sine of the angle also increases.

Step 2: Maximum Possible Value of Sum

Given that each angle \( \alpha, \beta, \gamma < \frac{\pi}{2} \), their sum \( \alpha + \beta + \gamma \) must be less than \( \frac{3\pi}{2} \). However, we are particularly interested in the case where this sum is less than \( \pi \), where the sine function is still increasing.

Step 3: Sine of Sum of Angles

For any three positive angles whose sum is \( \alpha + \beta + \gamma \), note that the maximum value of \( \sin(\alpha + \beta + \gamma) \) can only be achieved when the sum of these angles approaches \( \pi \). In that case, \( \sin(\alpha + \beta + \gamma) \leq 1 \).

Step 4: Sum of Sines of Individual Angles

Since \( \sin\alpha, \sin\beta, \sin\gamma \) are all positive (as they lie between 0 and 1 for angles in the first quadrant), their sum \( \sin\alpha + \sin\beta + \sin\gamma \) must be greater than 1.

Step 5: Comparison of Fraction

The fraction \( \frac{\sin(\alpha + \beta + \gamma)}{\sin\alpha + \sin\beta + \sin\gamma} \) must be less than 1 because \( \sin(\alpha + \beta + \gamma) \), although it can be at most 1, is divided by a sum that is definitely greater than 1. Thus, the fraction is always less than 1.

Key Concepts

Sine Function

The sine function is one of the fundamental components in trigonometry. It relates the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse. For an angle \( \theta \), the sine is denoted as \( \sin \theta \).

• Sine is a periodic function with a period of \( 2\pi \).
• It oscillates between -1 and 1.
• At \( 0, \pi, \) and \( 2\pi \), its value is 0, while at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), it reaches -1 and 1 respectively.

The sine function is essential when analyzing wave patterns and oscillations. It becomes particularly useful in various applications ranging from physics to engineering. Understanding the behavior of this function, especially its increasing and decreasing intervals, helps solve many trigonometric problems effectively. When looking at angles between 0 and \( \frac{\pi}{2} \) (first quadrant), the sine function increases from 0 to 1. This property is crucial when identifying the behavior of angles and their sine values within this quadrant.

Angles in First Quadrant

In trigonometry, knowing the quadrant in which an angle lies is vital for computing trigonometric functions. The first quadrant of the coordinate system is where both x and y coordinates are positive.

• It covers angles from \( 0 \) to \( \frac{\pi}{2} \).
• In this region, the sine, cosine, and tangent of angles are all positive.

Understanding angles in the first quadrant is important for several reasons:- **Sine Function:** Here, the sine function increases steadily from 0 to 1.- **Simple Calculations:** Calculating trigonometric functions in this range often involves fewer complexities since the values are non-negative and easy to visualize.For instance, if \( \alpha, \beta, \gamma \) fall within this quadrant, their respective sine values are always positive and less than or equal to 1. This quality simplifies many trigonometric evaluations, including summing multiple angle sines, as was needed in the original exercise.

Sum of Angles

The sum of angles is a fundamental concept in geometry and trigonometry. It often appears in problems where angles need to be combined to analyze their properties. When considering multiple angles \( \alpha, \beta, \gamma \) each residing in the first quadrant, the sum \( \alpha + \beta + \gamma \) plays a crucial role in understanding the behavior of their sine values.1. **Maximum Value:** For angles in the interval \( \left(0, \frac{\pi}{2}\right) \), the maximum value occurs when angles approach \( \pi \), as the sine function starts decreasing past this limit.2. **Relation to Sine:** The sine of the sum of these angles \( \, \sin(\alpha + \beta + \gamma) \, \) can't exceed 1 because the greatest point of the sine function is at 1.By comparing the sine of a combined angle sum to the sum of individual sines, we can determine important comparative relationships, such as identifying when one is greater or smaller than another. This principle was central to the exercise, showing that the numerator \( \sin(\alpha + \beta + \gamma) \) would always be less than the denominator \( \sin\alpha + \sin\beta + \sin\gamma \) in a specific range, leading to a fraction less than 1, thereby solving the problem.