Problem 17 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find the greatest and least values of \(\cos A \cos B\) when \(A+B=90^{\circ}\).

Step-by-Step Solution

Answer

The greatest value of \(\cos A \cos B\) is \(\frac{1}{2}\) and the least value is \(-\frac{1}{2}\) when \(A + B = 90^{\circ}\).

Step 1: Use Trigonometry Identity

We know that \( \cos B = \sin(90^{\circ} - A) = \sin A \). Substitute this into the function \(\cos A \cos B\) to get \(\cos A \sin A\).

Step 2: Convert to Single Trigonometric Ratio

Express \(\cos A \sin A\) in terms of a single trigonometric function. Using the identity \(2 \sin A \cos A = \sin 2A\), we can rewrite \(\cos A \sin A\) as \(\frac{1}{2} \sin 2A \).

Step 3: Evaluate Maximum and Minimum

The maximum and minimum values of the sine function occur at \(1\) and \(-1\) respectively. Since, \(\frac{1}{2} \sin 2A \) is just a rescaling of the sine function, the maximum and minimum values of \(\frac{1}{2} \sin 2A \) would be \(\frac{1}{2}\) and \(-\frac{1}{2}\) respectively.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables for which the expressions are defined. They serve as crucial tools in simplifying complex trigonometric equations and transforming expressions so that they can be computed more easily. In the context of the given exercise, we used a specific trigonometric identity:

• The co-function identity: \[\cos B = \sin(90^{\circ} - A)\]By substituting, we transformed \(\cos A \cos B\) into \(\cos A \sin A\). This change helps to simplify and better analyze the trigonometric expression.

Another vital identity used is:

• The double angle identity:\[2\sin A \cos A = \sin 2A\]This identity allows us to express the product of sine and cosine as a single sine function involving a double angle. Such transformations are often crucial when trying to maximize or minimize expressions as it changes the form into something more familiar, like the sine function, where we know the behavior well.

Trigonometric Functions

Trigonometric functions are fundamental in studying angles and their relationships to circles and right triangles. They include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), among others. These functions often describe how angles relate to the ratios of sides in right triangles. In our problem, we used:

• Cosine function: Initially, \( \cos A \) describes the cosine of angle \( A \), offering the adjacent to hypotenuse ratio in a right triangle.
• Sine function: \( \sin A \) is the sine of angle \( A \), describing the opposite to hypotenuse ratio. These functions' interplay helps to simplify solutions when angles are related, as with \( A+B=90^{\circ}\).

Understanding trigonometric functions is essential because different functions have different properties, such as ranges and periodicity. For example, sine and cosine range from -1 to 1, and those properties are integral in determining the maximum and minimum values in trigonometric scenarios.

Angle Sum and Difference Identities

Angle sum and difference identities are powerful tools used in trigonometry to find exact values of trigonometric functions for specific angles. These identities describe the trigonometric functions of sums or differences of angles in terms of the functions of the original angles. Presented in the exercise was:

• Sum identity for sine: \[\sin(A + B) = \sin A \cos B + \cos A \sin B\]This explains how sine can be expanded when dealing with the sum of two angles.

In our context, the angle sum identity helps by providing the relationship \( A+B=90^{\circ} \). From this equation, it becomes directly useful to transform one trigonometric function into another, assisting in simplifying calculations when there are angle constraints involved. Specifically, knowing \( A + B = 90^{\circ} \), thereby turning \( \cos B \) into \( \sin A \), a key step in simplifying \( \cos A \cos B \) to achieve our goal.