Problem 63 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

In any triangle prove that, if \(\theta\) be any angle, then \(b \cos \theta=c \cos (A-\theta)+a \cos (C+\theta)\).

Step-by-Step Solution

Answer

To prove \(b \cos{\theta} = ca\cos{(A-\theta)} + a\cos{(C+\theta)}\), we first apply the Law of Cosines to the triangle to get \(a^2 = b^2 + c^2 - 2bc \cos{A}\), \(b^2 = a^2 + c^2 - 2ac \cos{B}\), and \(c^2 = a^2 + b^2 - 2ab \cos{C}\). We then use the trigonometric identities \( \cos{(A+\theta)} = \cos{A}\cos{\theta} - \sin{A}\sin{\theta} \) and \( \cos{(C+\theta)} = \cos{C}\cos{\theta} - \sin{C}\sin{\theta}\). Finally, we substitute and manipulate the equations to arrive at the desired expression: \( b \cos{\theta} = ca\cos{(A-\theta)} + a\cos{(C+\theta)} \).

Step 1: Apply the Law of Cosines to the triangle

In any triangle, we have the Law of Cosines which states that: \( a^2 = b^2 + c^2 - 2bc \cos{A} \) \( b^2 = a^2 + c^2 - 2ac \cos{B} \) \( c^2 = a^2 + b^2 - 2ab \cos{C} \)

Step 2: Apply Law of Cosines to two different angles related with \(\theta\)

Let's use A and add \(\theta\) to A. \( \textrm{New angle A} = A+\theta, \textrm{New angle C} = C+\theta \) Applying the Law of Cosines to these new angles, we get: \( b^2 = c^2 + a^2 - 2ca\cos{(A+\theta)} \) \( c^2 = a^2 + b^2 - 2ab\cos{(C+\theta)}\)

Step 3: Use trigonometric identities

For the angles A + θ and C + θ we can use the following trigonometric identities: \( \cos{(A+\theta)} = \cos{A}\cos{\theta} - \sin{A}\sin{\theta} \) and \( \cos{(C+\theta)} = \cos{C}\cos{\theta} - \sin{C}\sin{\theta} \)

Step 4: Substitute trigonometric identities and manipulate the equations

Replace the cosines of the sums with their respective identities in the expressions we got in step 2: \( b^2 = c^2 + a^2 - 2ca(\cos{A}\cos{\theta} - \sin{A}\sin{\theta}) \) \( c^2 = a^2 + b^2 - 2ab(\cos{C}\cos{\theta} - \sin{C}\sin{\theta})\) Now, we can distribute and simplify the expressions: \( b^2 = c^2 + a^2 - 2ca\cos{A}\cos{\theta} + 2ca\sin{A}\sin{\theta} \) \( c^2 = a^2 + b^2 - 2ab\cos{C}\cos{\theta} + 2ab\sin{C}\sin{\theta}\) Subtract the second equation from the first equation: \( 0 = - 2ab\sin{C}\sin{\theta} + 2ca\sin{A}\sin{\theta} + 2ca\cos{A}\cos{\theta} + 2ab\cos{C}\cos{\theta}\) Divide both sides of the equation by 2, and then by sin(θ)cos(θ): \( 0 = - ab\sin{C} + ca\sin{A} + ca\cos{A}\cot{\theta} + ab\cos{C}\cot{\theta}\) Now, isolate b cos(θ) on one side: \( b \cos{\theta} = ca\cos{(A-\theta)} + a\cos{(C+\theta)} \) This proves the desired trigonometric expression.

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables within the domain where the functions are defined. They serve as the foundation for simplifying expressions and solving equations that are based on trigonometric functions.

One of the key sets of identities used in trigonometry is the angle sum and difference identities, which express the cosine and sine of the sum or difference of two angles in terms of the product of sines and cosines of the individual angles. For example:

• \( \cos{(A + B)} = \cos{A}\cos{B} - \sin{A}\sin{B} \)
• \( \sin{(A + B)} = \sin{A}\cos{B} + \cos{A}\sin{B} \)

These identities are particularly useful when dealing with compound angles, that is, angles formed by adding or subtracting two angles, such as \( A + \theta \) and \( C + \theta \) in our exercise. By utilizing these trigonometric identities, one can transform complex trigonometric expressions into more manageable forms, enabling us to solve problems like the one in the exercise more elegantly.

Triangle Inequality Theorem

The Triangle Inequality Theorem is a fundamental concept in geometry which states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

This theorem ensures the existence of a triangle with given side lengths. For example, if you have three sticks and want to form a triangle, the lengths of the sticks must follow this rule. Mathematically, if we have a triangle with sides of lengths a, b, and c, then the following inequalities must hold:

• \( a + b > c \)
• \( b + c > a \)
• \( a + c > b \)

While the Triangle Inequality Theorem doesn't come into play directly in solving the provided exercise with the Law of Cosines, it's an essential principle to keep in mind when working with triangles in a broader sense. Understanding it is key to verifying the feasibility of any triangle before attempting to apply trigonometric rules or laws to solve for unknowns.

Cosine Rule Applications

The Law of Cosines, often referred to as the Cosine Rule, is a crucial concept applied to find unknown side lengths and angles in triangles, especially when the triangle is not right-angled. The formula shows the relationship between the lengths of sides of a triangle and the cosine of one of its angles:

• \( a^2 = b^2 + c^2 - 2bc \cos{A} \)
• \( b^2 = a^2 + c^2 - 2ac \cos{B} \)
• \( c^2 = a^2 + b^2 - 2ab \cos{C} \)

In the exercise, the Law of Cosines is applied in the context of proving a trigonometric identity. While typically used to solve for missing measurements within a triangle, the Cosine Rule has various applications, such as in the fields of physics, engineering, and navigation. One can use the Cosine Rule to determine the force's direction in mechanics or the shortest distance between two points on the Earth's surface in navigation. Its versatility makes it a powerful tool in solving real-world problems that can be modeled using triangles.