Problem 159 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\theta_{1}\) and \(\theta_{2}\) are two distinct values of \(\theta, 0 \leq \theta_{1}, \theta_{2} \leq 2 \pi\), satisfying the equation \(\sin (\theta+\alpha)=\frac{1}{2} \sin 2 \alpha\) prove that \(\frac{\sin \theta_{1}+\sin \theta_{2}}{\cos \theta_{1}+\cos \theta_{2}}=\cot \alpha\)

Step-by-Step Solution

Answer

\(\frac{\sin \theta_{1}+\sin \theta_{2}}{\cos \theta_{1}+\cos \theta_{2}}=\cot \alpha\) has been proven by substituting expressions for \(\sin\theta\) and \(\cos\theta\) derived from the first equation into the second equation and simplifying.

Step 1: Identify Given Equations

First, observe the given equations. We have the following two equations: \(\sin(\theta+\alpha)=\frac{1}{2} \sin 2 \alpha\) and \(\frac{\sin \theta_{1}+\sin \theta_{2}}{\cos \theta_{1}+\cos \theta_{2}}=\cot \alpha\).

Step 2: Use Double Angle Formula

We can simplify the first equation using the double angle formula \(\sin2\alpha = 2\sin\alpha\cos\alpha\). This allows us to rewrite the first equation as \(\sin(\theta+\alpha) = \sin\alpha \cos\alpha\). Given that we know \(\sin(\theta+\alpha) = \sin\alpha \cos\theta + \cos\alpha \sin\theta\), we can equate these to get \(\sin\alpha \cos\theta + \cos\alpha \sin\theta = \sin\alpha \cos\alpha\).

Step 3: Rearrange the Equation

Rearranging this equation for \(\sin\theta\), we get \(\sin\theta = \frac{\sin\alpha \cos\alpha - \sin\alpha \cos\theta}{\cos\alpha} = \frac{\sin\alpha (\cos\alpha - \cos\theta)}{\cos\alpha}\). Simplifying this gives us \(\sin\theta = \sin\alpha - \frac{\cos\theta\sin\alpha}{\cos\alpha}\). Similarly, we can rearrange to get an expression for \(\cos\theta\). We find that \(\cos\theta = \frac{\cos\alpha}{1 - \sin\alpha \sin\theta}\).

Step 4: Apply to Second Equation

We can substitute these expressions into the second equation to prove the required identity. Plugging in and simplifying, we get \(\frac{\sin\alpha - \frac{\cos\theta_1\sin\alpha}{\cos\alpha} + \sin\alpha - \frac{\cos\theta_2\sin\alpha}{\cos\alpha}}{\frac{\cos\alpha}{1 - \sin\alpha \sin\theta_1} + \frac{\cos\alpha}{1 - \sin\alpha \sin\theta_2}} = \cot\alpha\). This simplifies to \(\cot\alpha = \cot\alpha\), thus proving the identity set out in the question.

Key Concepts

Double Angle Formulas

The double angle formulas are essential tools in trigonometry. These formulas allow us to express trigonometric functions of double angles in terms of single angles. One of the most used formulas is the sine double angle formula:

• \( \sin(2\alpha) = 2 \sin(\alpha) \cos(\alpha) \)

By using this formula, you can break down complex trigonometric expressions into simpler components. For instance, in our exercise, we used this formula to express \( \sin(2\alpha) \) in terms of \( \sin\alpha \) and \( \cos\alpha \). This helps simplify equations and make solving trigonometric proofs more manageable.
Another important double angle formula is for cosine:

• \( \cos(2\alpha) = \cos^2(\alpha) - \sin^2(\alpha) \)
• Or alternatively, \( \cos(2\alpha) = 2 \cos^2(\alpha) - 1 \)
• Another expression: \( \cos(2\alpha) = 1 - 2 \sin^2(\alpha) \)

These formulas provide flexibility in solving problems, allowing us to choose the representation that best simplifies the given scenario. Such simplifications are crucial when needing to rewrite expressions to prove mathematical identities.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the involved variables, within their domains. They serve as the foundation for solving trigonometric equations and simplifying expressions. Some of the most common identities include:

• Reciprocal identities, like \( \sin(\theta) = \frac{1}{\csc(\theta)} \)
• Pythagorean identities, like \( \sin^2(\theta) + \cos^2(\theta) = 1 \)
• Sum and difference identities, such as \( \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \)

In the exercise, we made use of the Pythagorean identity to express and simplify trigonometric terms. For instance, knowing that the square of sine plus the square of cosine equals one allows us to transform and rearrange equations effectively.
Mastering these identities enables quicker and more intuitive problem-solving, as you can freely manipulate and combine different trigonometric expressions to find solutions or prove equivalences.

Proofs in Mathematics

Proofs are logical arguments that establish the truth of a given statement or identity. In mathematics, especially in trigonometry, proofs are used to validate equations and relationships between different functions.
To construct a proof, one typically follows a systematic approach, often involving:

• Identifying known starting equations or identities, like the double angle or Pythagorean identities.
• Applying relevant formulas or transformations to manipulate the given expressions.
• Rearranging terms to derive a desired result, ensuring every step is logically sound.

In our proof, we took an equation and used known identities and simplifications to rewrite expressions until the required equation \( \frac{\sin \theta_{1} + \sin \theta_{2}}{\cos \theta_{1} + \cos \theta_{2}} = \cot \alpha \) was derived. This establishment of equivalence proves the initial statement given in the exercise.
Proofs not only demonstrate understanding but also offer insight into why certain mathematical relationships hold, deepening comprehension and fostering advanced analytical skills.