Problem 29 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

In a \(\Delta A B C, a^{2} \cos ^{2} A=b^{2}+c^{2}\), then (a) \(A<\frac{\pi}{4}\) (b) \(\frac{\pi}{4}\frac{\pi}{2}\) (d) \(A=\frac{\pi}{2}\)

Step-by-Step Solution

Answer

Based on the solution steps, angle A should be equal to \(\frac{\pi}{2}\). So, the correct choice is option (d) \(A=\frac{\pi}{2}\)

Step 1: Simplify the equation

Start by rearranging the given equation \(a^{2} \cos ^{2} A=b^{2}+c^{2}\) to isolate \(\cos ^{2} A\). This is done by dividing both sides by \(a^{2}\) , getting us \(\cos ^{2} A = \frac{b^{2}+c^{2}}{a^{2}}\).

Step 2: Analysis of cosine function

Recall the properties of cos function, specifically that the maximum value of cos and, consequently, \(\cos ^{2} A\), can take is 1.

Step 3: Comparing values

Given \( \cos ^{2} A = \frac{b^{2}+c^{2}}{a^{2}} \), knowing that \(\cos ^{2} A\) cannot be more than 1, we can infer that \(\frac{b^{2}+c^{2}}{a^{2}}\) cannot exceed 1. This implies that \(b^{2} + c^{2} \leq a^{2}\). This is the property of a right-angled triangle (Pythagoras theorem), where the square of the hypotenuse (a) is equal to the sum of squares of the other two sides (b and c). Hence, the triangle is a right-angled triangle.

Step 4: Deciding angle range

As we know the triangle is right-angled, the angle A which is opposite to side a (Hypotenuse) must be \(\frac{\pi}{2}\).

Key Concepts

Cosine Function Properties

The cosine function is a fundamental concept in trigonometry, crucial in analyzing angles and triangles. It represents the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Some key properties, essential for understanding its behavior, are:

• Range of Cosine: The cosine of an angle varies only between -1 and 1. This is because the adjacent side can never be larger than the hypotenuse.
• Even Function: Cosine is an even function. This means \( \cos(-x) = \cos(x) \).
• Periodic Nature: Cosine has a period of \( 2\pi \). Every \( 2\pi \) intervals repeat the same cosine values.

When you take the square of cosine, such as \( \cos^{2} A \), it implies the range shifts between 0 and 1. This can confirm triangle properties, essential for determining the type of triangle or angle range.

Right-angled Triangle Theorem

The right-angled triangle theorem, famously known as the Pythagorean theorem, plays a pivotal role in evaluating triangle properties using the cosine function. It states that in a right triangle:

• The square of the hypotenuse \( a \) is equal to the sum of the squares of the other two sides \( b \) and \( c \).

Given as \( a^{2} = b^{2} + c^{2} \), this theorem is vital in proving a triangle is right-angled when given side lengths. In certain problems, like the one discussed, if \( \cos^2 A \) results in a relationship \( b^{2} + c^{2} \leq a^{2} \), it signifies a right angle, which confirms side \( a \) is indeed the hypotenuse with angle \( A \) being \( \frac{\pi}{2} \). This emphasizes the importance of angles and sides in categorizing triangles.

Angle Range Analysis

Understanding the range of an angle based on given trigonometric conditions is crucial. For the cosine squared value derived from the original exercise, the analysis helps in determining the possible ranges of angle \( A \).The hypothesis \( \frac{b^{2} + c^{2}}{a^{2}} \leq 1 \) indicated a specific angle property by linking back to the cosine's maximum value of 1. When it's stated that \( A = \frac{\pi}{2} \), it presents a broader view:

• Less than \( \frac{\pi}{2} \): Implies a possibly obtuse or right angle based on the sum of squares.
• Greater than \( \frac{\pi}{2} \): Would defy the current equation scenario since it doesn't align with the cosine function's properties.
• Exactly \( \frac{\pi}{2} \): Matches a right angle, further conforming to Pythagorean principles.

This thorough analysis allows us to ascertain the angle accurately through mathematical reasoning.