Problem 166 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \tan \theta=\frac{b}{a}, \text { prove that } a \cos 2 \theta+b \sin 2 \theta=a \text { . } $$

Step-by-Step Solution

Answer

The given statement \(a \cos 2\theta + b\sin 2\theta = a\) is verified and hence proven.

Step 1: Replace \(\tan \theta\) with \(b/a\)

According to the problem, \(\tan \theta = b/a\). This means that \(b = a \tan \theta\), and \(a = b / \tan \theta\).

Step 2: Express \(\cos 2\theta\) and \(\sin 2\theta\) in terms of \(\tan \theta\)

The double angle formulas for cosine and sine are \(\cos 2\theta = 1 - 2 \tan^2\theta / (1+\tan^2\theta)\), and \(\sin 2\theta = 2 \tan\theta / (1 + \tan^2\theta)\). Substituting \(b/a\) for \(\tan\theta\) we get \(\cos 2\theta = 1 - 2b^2 / (a^2 + b^2)\) and \(\sin 2\theta = 2ab / (a^2 + b^2)\).

Step 3: Substitute and Simplify

Substitute these expressions into \(a \cos 2\theta + b \sin 2\theta = a\). We have \(a[1-2b^2/(a^2+b^2)] + b[2ab/(a^2+b^2)] = a\). Simplifying this gives \(a = a\), which is true.

Key Concepts

Double Angle Formulas

Double angle formulas are fundamental tools in trigonometry used to simplify the computation of trigonometric functions at twice a given angle. Specifically, they help to convert expressions involving \( \sin(2\theta) \) or \( \cos(2\theta) \) into forms that are dependent on \( \sin(\theta) \) or \( \cos(\theta) \) alone. The main double angle formulas are:

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

In solving trigonometric problems, especially those involving quadratic relationships, these formulas are invaluable as they allow the conversion of complex expressions into simpler ones. For instance, by transforming \( \cos(2\theta) \) and \( \sin(2\theta) \) using double angle formulas as shown in the exercise, we can easily facilitate proofs involving these functions.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent, are relationships between the angles and sides of a triangle or the angles and ratios of a unit circle. They are the backbone of trigonometry and have applications in various fields like physics, engineering, and architecture.

• \( \sin(\theta) \) is the ratio of the opposite side to the hypotenuse.
• \( \cos(\theta) \) is the ratio of the adjacent side to the hypotenuse.
• \( \tan(\theta) \) is the ratio of the opposite to the adjacent side or \( \sin(\theta) / \cos(\theta) \).

Their reciprocal functions, cosecant, secant, and cotangent, are equally vital. Trigonometric identities, like the one being proven in the exercise, demonstrate constant relationships between these functions regardless of the value of the angle, thus providing valuable tools for solving and simplifying mathematical expressions and equations.

Proof Techniques in Calculus

Proof techniques in calculus are structured approaches to validate mathematical propositions and theorems, which include limit theorems, derivatives, integrals, and the properties of trigonometric functions. Common techniques include:

• Direct proof: involves deducing the truth of a statement from known facts through a series of logical steps.
• Proof by contradiction: assumes that the statement to be proved is false and derives a contradiction from this false assumption.
• Induction: good for proving statements dependent on integers, starting with a base case and proving that if the statement holds for an integer \( n \), it also holds for \( n+1 \).

When proving trigonometric identities, like in our example, understanding how to manipulate expressions using trigonometric functions and their properties becomes crucial. Through these techniques, one can demonstrate the equality of two mathematical expressions, establish the validity of a theorem, or explore the nature of a function.