Problem 28 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

The minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma\), where \(\alpha, \beta, \gamma\) are real +ve angles satisfying \(\alpha+\beta+\gamma=\pi\), is (a) \(+\mathrm{ve} \quad\) (b) \(-\mathrm{ve}\) (c) 0 (d) \(-3\)

Step-by-Step Solution

Answer

The minimum value of the expression is -3 (option d)

Step 1: Understand the Problem

The problem asks for the minimum value of the expression \(\sin \alpha+\sin \beta+\sin \gamma\), where \(\alpha, \beta, \gamma\) are real positive angles such that \(\alpha+\beta+\gamma=\pi\), or 180 degrees.

Step 2: Visualize the Problem

Since we are talking about sine function, we know that the sine function varies between -1 and +1. Therefore, irrespective of their actual value, the minimum value for each \(\sin \alpha, \sin \beta, \sin \gamma\) can only be -1.

Step 3: Solution

Given that the sum \(\alpha+\beta+\gamma= \pi\), then their corresponding sines sum to \(\sin \alpha+\sin \beta+\sin \gamma\). Since the minimum possible value for each \(sin \alpha, sin \beta, sin \gamma \) is '-1'. The minimum possible sum would be \('-1 - 1 - 1\) which equals to '-3'.

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry, defining the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. This function is periodic and oscillates between -1 and +1.

Importantly, for angles in a right-angled triangle, the sine of an angle greater than 90 degrees and less than 270 degrees is negative, whereas it is positive for angles less than 90 degrees or greater than 270 degrees. Due to its periodic nature, the sine function repeats its values every 360 degrees, or equivalently, \(2\pi\) radians.

In the context of the given problem, understanding the sine function's range and periods is crucial for determining the minimum value of the trigonometric expression involved.

Angle Sum Identity

The angle sum identity is an essential principle in trigonometry that provides a way to decompose the sine or cosine of the sum of two angles. Specifically, the identity for sine is \( \sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta \).

This allows us to understand how the sine of a compound angle can be expressed in terms of the sines and cosines of its individual components. For the exercise in question, while the angle sum identity itself is not directly used, it underpins the broader understanding of how trigonometric functions behave when dealing with sums of angles, which is related to the condition \( \alpha + \beta + \gamma = \pi \).

Properties of Trigonometric Functions

Trigonometric functions come with a set of properties that define their behavior. One important property is that trigonometric functions are bounded, meaning that they have maximum and minimum values that they do not exceed. For sine and cosine functions, these bounds are -1 and +1.

Another crucial property is that they are periodic, repeating their values after a certain interval, which for sine and cosine is \(2\pi\) radians or 360 degrees. Additionally, symmetrical properties provide insights into the function values at different quadrants of the unit circle.

When considering the original problem of finding the minimum value of a sum of sine functions, these properties guide us to understand that the sum will be minimized when each individual sine function is at its minimum value, which is when the angle corresponds to 270 degrees or \( \frac{3\pi}{2} \) radians, where the sine function value is -1.