Problem 68 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(\mathrm{A}\) and \(\mathrm{B}\) denote the statements A: \(\cos \alpha+\cos \beta+\cos \lambda=0\) B: \(\sin \alpha+\sin \beta+\sin \lambda=0\) If \(\cos (\beta-\lambda)+\cos (\beta-\alpha)+\cos (\alpha-\beta)=-\frac{3}{2}\), then (A) A is true and B is false (B) A is false and \(\mathrm{B}\) is true (C) both \(\mathrm{A}\) and \(\mathrm{B}\) are true (D) both A and B are false

Step-by-Step Solution

Answer

(C) both A and B are true.

Step 1: Analyze given cosine equation

We start with the equation \( \cos (\beta - \lambda) + \cos (\beta - \alpha) + \cos (\alpha - \beta) = -\frac{3}{2} \). This suggests that the angular differences have a particular relationship that might help us deduce something about the individual angles.

Step 2: Simplify using cosine identity

We use the formula \( \cos(A - B) = \cos A \cos B + \sin A \sin B \) and apply it to all terms: \( \cos(\beta - \lambda), \cos(\beta - \alpha), \text{ and } \cos(\alpha - \beta). \) This is equivalent to having the identity forms lying on axes of an equilateral triangle. It only holds if angles are \(120^{\circ}, -120^{\circ}, \text{and } 0^{\circ}. \)

Step 3: Verify cosine condition

If the angles are equilateral as described above, the sum of their respective cosines should cancel out. Thus \( \cos(\alpha) + \cos(\beta) + \cos(\lambda) = 0 \), satisfying statement A.

Step 4: Verify sine condition

Similarly check the sine values. With equilateral angles, the sine sums also cancel \( \sin(\alpha) + \sin(\beta) + \sin(\lambda) = 0 \), thereby satisfying statement B as well.

Step 5: Determine truth values of A and B

Given both verifications, both conditions A and B are true according to their equations.

Key Concepts

Cosine Law

The Cosine Law is a fundamental principle in trigonometry, particularly useful in solving triangles. It connects the sides of a triangle to the cosine of one of its angles. In any triangle with sides denoted as \(a\), \(b\), and \(c\), and angle \(C\) opposite side \(c\), the formula is:
\[c^2 = a^2 + b^2 - 2ab \cdot \cos C\]This law helps in situations where you know:

• Two sides and the included angle, allowing you to find the third side.
• All three sides, enabling you to calculate any angle.

It's especially handy for non-right triangles where the Pythagorean theorem doesn't apply. In the exercise, the cosine of angular differences uses the identity \(\cos(A - B) = \cos A \cdot \cos B + \sin A \cdot \sin B\) to analyze and simplify conditions. Understanding how these relationships work is key to finding solutions in trigonometric problems.

Sine Rule

The Sine Rule, or Sine Law, is another valuable tool in trigonometry for solving triangles. It relates the sides of a triangle to the sines of its angles. The formula is:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This rule is particularly useful when:

• You need to know a side opposite a given angle, and you have another side and angle known.
• You need to find an angle given two sides and an angle not sandwiched between those sides.

In the provided solution, verifying that the sum of the sines of equilateral angles equals zero illustrates that the Sine Rule effectively operates even under special conditions like equilateral triangles. Such insights help deepen your understanding of trigonometric identities.

Equilateral Triangle Angles

Equilateral triangles are intriguing in trigonometry due to their symmetry. All sides and angles are equal. This special property means each angle in an equilateral triangle is \(60^{\circ}\). However, in the exercise, we dive into a less conventional perspective where angles behave like \(120^{\circ}, -120^{\circ}, \) and \(0^{\circ}\). Here's why this works:

• Adding these angles maintains their circular nature, adding up to a full circle or a rearranged congruency.
• The cosine and sine relationships here simplify due to repeated and symmetric behavior, like canceling out to zero.

These specific angles are important when analyzing geometric patterns or cyclic behavior. Recognizing these special configurations can expand your understanding of how triangles can be represented in different trigonometric scenarios.