Problem 34 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(a^{2}-2 a b \cos \left(60^{\circ}+C\right)=c^{2}-2 b c \cos \left(60^{\circ}+A\right)\)

Step-by-Step Solution

Answer

The equation relating the sides and angles of a triangle using cosine and sine functions is \( a^{2} + bc\cos(A) - bc\sqrt{3}\sin(A) = c^{2} + ab\cos(C) - ab\sqrt{3}\sin(C) \). This relationship could be useful in solving for unknown sides or angles in a triangle.

Step 1: Analyse the equations

From the equation, we are given two expressions that comprise of the law of cosines. Our task is to simplify the expressions, and hopefully find a connection between the two.

Step 2: Substitute cosine sum angles formula

Recall, \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \). Hence, substituting the value of \( \cos(60 + C) \) and \( \cos(60 + A) \) into the equations, we have \( a^{2} - 2ab(\cos(60)\cos(C) - \sin(60)\sin(C)) = c^{2} - 2bc(\cos(60)\cos(A) - \sin(60)\sin(A)) \)

Step 3: Simplify the equation

We know the values of \( \cos(60) = 1/2 \) and \( \sin(60) = \sqrt{3}/2 \). Substituting these into the equation and simplifying the like terms we get \( a^{2} - ab\cos(C) + ab\sqrt{3}\sin(C) = c^{2} - bc\cos(A) + bc\sqrt{3}\sin(A) \)

Step 4: Rearrange the equation

Rearrange the equation such that we have all terms in \( a \) on one side and all terms in \( c \) on the other: \( a^{2} + bc\cos(A) - bc\sqrt{3}\sin(A) = c^{2} + ab\cos(C) - ab\sqrt{3}\sin(C) \)

Step 5: Interpret the result

The resulting equation relates the sides and angles of a triangle in terms of cosine and sine functions. This relation could be useful in solving for the unknowns.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental equations involving trigonometric functions. They hold true for all angle measures. These identities simplify complex trigonometric expressions by allowing us to substitute and rearrange parts of an equation effectively.

A basic yet pivotal identity is the Pythagorean Identity:

• \(\sin^2(\theta) + \cos^2(\theta) = 1\)

This can act as the basis for many other transformations and deductions. Knowing fundamental identities like this can make it easier to simplify equations during problem-solving.

In our exercise, we used the cosine sum identity, which reads:

• \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\)

By applying the cosine sum identity, trigonometric expressions involving sums of angles become more manageable, and we can break down these expressions into simpler components. Understanding and applying these identities is crucial for solving trigonometric problems efficiently.

Angle Sum Formula

The Angle Sum Formula is an essential concept in trigonometry. It helps us compute the trigonometric functions of angles that are expressed as the sum of two other angles. This formula expands a trigonometric function of a sum into a more simple expression based on the individual angles.

For cosine, this formula is given by:

• \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\)

In practice, the Angle Sum Formula can transform expressions like \(\cos(60^{\circ}+C)\) seen in our exercise. Here, both \(a\) and \(b\) represent angles, such as \(60^{\circ}\) and another angle like \(C\).

Implementing these formulas breaks down complex equations effectively. This transformation is crucial in proving identities or reducing equations to simpler forms, making calculations easier and maintaining the focus on understanding relationships between the angles themselves.

Triangle Properties

Understanding the properties of triangles is vital to solving problems involving relationships between sides and angles. In particular, a fundamental tool in trigonometry and geometry is the Law of Cosines.

The Law of Cosines is useful for calculating the unknown components of a triangle when given sufficient information. It’s formulated as:

• \(c^2 = a^2 + b^2 - 2ab\cos(C)\)

This law relates the lengths of the sides of a triangle to the cosine of one of its angles. In our exercise, the Law of Cosines assists in exploring the connections between the sides and angles, simplifying the expression.

Grasping these properties not only aids in solving mathematical issues but also enlightens us about the intrinsic harmony in geometric shapes. This enables us to predict and manipulate relationships in triangles comprehensively, whether solving for an angle, a side, or understanding the spatial relationships within.