Problem 26 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In any \(\Delta A B C, \Pi\left(\frac{\sin ^{2} A+\sin A+1}{\sin A}\right)\) is always greater than (a) 9 (b) 3 (c) 27 (d) 36

Step-by-Step Solution

Answer

The minimal value for the given expression is greater than 9. Therefore, the correct answer is (a) 9.

Step 1: Simplify the given expression

Notice that the given expression can be split and simplified. Divide each term of the numerator by \(\sin A\), thus the expression becomes: \(\Pi(\sin A + 1 + \csc A)\), where \(\csc A\) is the cosecant of A, defined as \(1/\sin A\).

Step 2: Apply trigonometric identity

Recall that cosecant of any angle in a right-angled triangle is always more than or equal to 1. Therefore, \(1 + \sin A + \csc A\) will certainly be greater than 1 + 1 + 1 = 3.

Step 3: Final calculation

The pi function multiplies its argument by pi \(\Pi(x) = \pi*x\). Given that our simplified expression was greater than 3, when it is multiplied by \(\pi\) it will be greater than \(3*\pi\). As \(\pi\) is approximately 3.1416, this value will be greater than 9.

Key Concepts

Cosecant Function

The cosecant function, often denoted as \( \csc \theta \), is a fundamental concept in trigonometry. It is the reciprocal of the sine function. For an angle \( A \), the cosecant is defined by the equation \( \csc A = \frac{1}{\sin A} \). This means it relates directly to the sine function:
\[ \csc A = \frac{1}{y} \] where \( y \) is the sine value of the angle \( A \).

Key Points to Remember:

• The cosecant function is undefined when \( \sin A = 0 \), as division by zero is undefined.
• In the context of a right-angled triangle, \( \csc A \) represents the ratio of the hypotenuse to the opposite side of angle \( A \).
• The range of the cosecant function is \((−\infty, -1] \cup [1, \infty)\) because it never lies between −1 and 1.

Understanding the cosecant function is crucial because it aids in solving trigonometric equations and manipulating expressions, as in the given problem where the cosecant helps simplify the expression\( \sin^2 A + \sin A + 1 \). By transforming the terms using \( \csc A \), simplification becomes intuitive and calculation becomes deeper into trigonometric structures.

Trigonometric Identities

Trigonometric identities are relations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equality are defined. These identities are foundational tools in simplifying trigonometric expressions and proving equations.

Some important trigonometric identities include:

• Pythagorean Identities, such as \( \sin^2 A + \cos^2 A = 1 \).
• Reciprocal Identities, like \( \csc A = \frac{1}{\sin A} \), which were used in this exercise's solution.
• Angle Sum and Difference Identities, such as \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \).

These identities help in transforming and simplifying expressions. For instance, using the reciprocal identity \(\csc A = \frac{1}{\sin A}\), the expression \(\sin^2 A + \sin A + 1\) was simplified by separating terms.

This simplification helps in recognizing inner structures of trigonometric equations, leading to efficient solving methods.

Triangle Inequality

The triangle inequality is a fundamental concept that applies to triangles, stating that the sum of the lengths of any two sides must always be greater than the length of the third side. This principle ensures that a valid triangle can actually be formed.

General Properties:

• For any triangle with sides \( a \), \( b \), and \( c \): \( a + b > c \), \( b + c > a \), \( a + c > b \).
• This inequality ensures triangle existence and constraints in geometric constructions.

In the context of trigonometric inequalities, similar concepts apply. Trigonometric functions may relate side ratios in triangle setups, forming inequalities that help in determining valid ranges for expressions. In this problem, after simplifying the initial expression to \( \sin A + 1 + \csc A \), knowing that \( \csc A \geq 1 \) assists in concluding that the expression exceeds a certain threshold—much like the triangle inequality ensuring the existence of triangle sides.

This analogy signals that whether in geometry or algebra, inequalities provide powerful tools to assert valid conclusions and solve for unknowns.