Question

If \(\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)\) then the value of \(\frac{\sin (\alpha+\beta+\gamma)}{\sin \alpha+\sin \beta+\sin \gamma}\) is (1) less than 1 (2) greater than 1 (3) equal to 2 (4) greater than 2

Short answer

(1) less than 1

Step-by-step solution

Understanding the Range of Angles

Given that \(\alpha, \beta, \gamma \in\left(0, \frac{\pi}{2}\right)\), all three angles are between 0 and \(\frac{\pi}{2}\) which means all angles are positive and less than \(\frac{\pi}{2}\)

Analyzing the Trigonometric Functions

Since \(\alpha, \beta, \gamma\) are in the first quadrant, \(\sin(\alpha)\), \(\sin(\beta)\), and \(\sin(\gamma)\) are all positive and less than 1. The sum of three positive numbers less than 1 does not exceed 3.

Simplifying the Expression

Consider the values of both numerator and denominator of the expression. The numerator \(\sin(\alpha + \beta + \gamma)\) is a trigonometric value bounded by \(-1\) and \(1\). Since \(\alpha, \beta, \gamma\) are in the first quadrant, the value lies between 0 and 1.

Comparing Numerator and Denominator

The denominator \(\sin(\alpha) + \sin(\beta) + \sin(\gamma)\) consists of three positive sine values, each less than 1. Therefore, their sum is also less than 3 and greater than 0. The numerator is between 0 and 1, while the denominator is between 0 and 3.

Evaluating the Fraction

Since the numerator (bounded by 0 and 1) is always less than or equal to the denominator (bounded by 0 and 3), the value of the fraction \(\frac{\sin(\alpha + \beta + \gamma)}{\sin \alpha + \sin \beta + \sin \gamma}\) is always less than 1.

Key concepts

Trigonometric Functions

In the context of the given exercise, we are dealing with the sine function. The sine function, denoted as \(\text{sin}(x)\), is one of the fundamental trigonometric functions. It associates any angle \(\theta\) with a value from -1 to 1. This value represents the y-coordinate of a point on the unit circle corresponding to that angle. For angles between 0 and \(\frac{\text{pi}}{2}\), the sine of these angles is positive and ranges from 0 to 1. Therefore, in our problem where \(\alpha, \beta, \gamma\) are all in the first quadrant \((0, \frac{\text{pi}}{2})\), we have \(\text{sin}(\alpha)\), \(\text{sin}(\beta)\), and \(\text{sin}(\gamma)\), all being positive, non-negative, and less than 1. Each of these trigonometric functions behaves nicely, as long as the angles remain confined to this range.

Angle Ranges

Understanding the ranges in which the angles lie is crucial. The problem states that \(\text{sin}(\alpha+\beta+\gamma)\) for \(\alpha, \beta, \gamma \in(0, \frac{\text{pi}}{2})\). Let's break this down. Angles in the range of 0 to \(\frac{\text{pi}}{2}\) are in the first quadrant of the unit circle. This range guarantees that the sine values of these angles are always between 0 and 1. These angles do not approach negative values, and do not hit zero. They remain in this positive and predictable region of the sine function. This also simply means that \alpha, \beta, and \gamma are all strictly positive acute angles. When these angles are added, their resulting sine value, \(\text{sin}(\alpha + \beta + \gamma)\), will also stay within the bounds of the first quadrant. This results in values of sine that are equally manageable and known to be less than or equal to 1.

Fraction Evaluation

In mathematics, evaluating fractions often involves understanding the behavior of the numerator and the denominator independently. In this problem, we are given the fraction \(\frac{\text{sin}(\alpha + \beta + \gamma)}{\text{sin}(\alpha) + \text{sin}(\beta) + \text{sin}(\gamma)}\). The numerator here, \(\text{sin}(\alpha + \beta + \gamma)\), is bounded to values between 0 and 1. This is a consequence of \(\alpha+\beta+\gamma\) also lying within the range \(0, \frac{\text{pi}}{2}\). The denominator is a bit more complex; it is the sum of three sine values each of which, as established before, is between 0 and 1. Therefore, the sum of these three values lies between 0 and 3. When comparing a numerator that does not exceed 1 to a denominator that can range from 0 up to almost 3, it's easy to see that the resulting fraction's value must be less than 1. This analysis tells us that the given fraction \(\frac{\text{sin}(\alpha + \beta + \gamma)}{\text{sin}(\alpha) + \text{sin}(\beta) + \text{sin}(\gamma)}\) cannot exceed 1, as the numerator is constrained to be equal or smaller than the denominator.