Question

\(\frac{\sin x}{\sin \left(\frac{x}{8}\right)}\) is equal to (a) \(8 \sin \left(\frac{x}{2}\right) \cos \left(\frac{x}{4}\right) \cos \left(\frac{x}{8}\right)\) (b) \(8 \cos \left(\frac{\mathrm{x}}{2}\right) \sin \left(\frac{\mathrm{x}}{4}\right) \cos \left(\frac{\mathrm{x}}{8}\right)\) (c) \(8 \cos \left(\frac{\mathrm{x}}{2}\right) \cos \left(\frac{\mathrm{x}}{4}\right) \cos \left(\frac{\mathrm{x}}{8}\right)\) (d) \(4 \cos \left(\frac{x}{2}\right) \cos \left(\frac{x}{4}\right) \cos \left(\frac{x}{8}\right)\)

Short answer

To find the equivalent expression for the given trigonometric expression \(\frac{\sin x}{\sin \left(\frac{x}{8}\right)}\), we used the Product-to-Sum formulas and double angle formulas to rewrite the given expression in a simpler form. After simplification, we found the equivalent expression to be: \(8 \cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\) Comparing with the given options, we found that this expression matches option (b).

Step-by-step solution

Rewrite the expression using Product-to-Sum formulas

We will rewrite \(\sin x\) as a product of trigonometric functions involving its half angles and quarter angles, using the Product-to-Sum formulas: \(\sin x = 2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)\), using the double angle formula, Then, we have: \(\sin x = 4\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) [\cos^2\left(\frac{x}{4}\right) - \sin^2\left(\frac{x}{4}\right)]\) Now, rewriting the expression \(\sin\left(\frac{x}{8}\right)\) using Product-to-Sum formula: \(\sin\left(\frac{x}{8}\right) = \frac{1}{2}[\sin\left(\frac{x}{4}\right) - \sin\left(\frac{x}{4} - \frac{x}{8}\right)]\) And, we have: \(\frac{\sin x}{\sin \left(\frac{x}{8}\right)} = \frac{4\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) [\cos^2\left(\frac{x}{4}\right) - \sin^2\left(\frac{x}{4}\right)]}{\frac{1}{2}[\sin\left(\frac{x}{4}\right) - \sin\left(\frac{x}{4} - \frac{x}{8}\right)]}\)

Simplify the expression

Now, we will simplify the expression by canceling the common factors: \(\frac{\sin x}{\sin \left(\frac{x}{8}\right)} = 8\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) [\cos^2\left(\frac{x}{4}\right) - \sin^2\left(\frac{x}{4}\right)] [\sin\left(\frac{x}{4}\right)]^{-1} [\sin\left(\frac{x}{4} - \frac{x}{8}\right)]^{-1}\) Now, using the double angle formula for angles \(\frac{x}{4}\) and \(\frac{x}{4} - \frac{x}{8}\), we get: \(\frac{\sin x}{\sin \left(\frac{x}{8}\right)} = 8\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right) [\cos^2\left(\frac{x}{4}\right) - \sin^2\left(\frac{x}{4}\right)] [\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{8}\right)]^{-1} [\sin\left(\frac{3x}{8}\right)\cos\left(\frac{3x}{8}\right)]^{-1}\) Now, cancel the common factors: \(\frac{\sin x}{\sin \left(\frac{x}{8}\right)} = 8 \cos\left(\frac{x}{2}\right)\sin\left(\frac{x}{4}\right)\cos\left(\frac{x}{8}\right)\) Comparing this result with the given options, we can see that it matches option (b). Therefore, the correct answer is: (b) \(8 \cos \left(\frac{\mathrm{x}}{2}\right) \sin\left(\frac{\mathrm{x}}{4}\right) \cos \left(\frac{\mathrm{x}}{8}\right)\).

Key concepts

Double Angle Formulas

Double angle formulas are key tools in trigonometry used to simplify expressions involving angles that are twice as large as a given angle. Common double angle formulas include:

\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \
\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \
\cos(2\theta) = 2\cos^2(\theta) - 1 \
\cos(2\theta) = 1 - 2\sin^2(\theta) \] By using these formulas, more complex trigonometric problems, such as the one from the IIT-JEE exam, can be broken down into simpler parts for easier calculation. Understanding how to apply the double angle formulas is crucial for students preparing for competitive exams like the IIT-JEE, where time management and problem-solving skills are tested.

Product-to-Sum Formulas

Product-to-sum formulas are another set of trigonometric identities that express products of sine and cosine functions as sums or differences of trigonometric functions. They are especially useful when dealing with products of trigonometric functions in an expression, making them integral for simplifying and solving trigonometric equations. The key formulas are:

\[ \sin A \cos B = \frac{1}{2}[\sin(A - B) + \sin(A + B)] \
\cos A \sin B = \frac{1}{2}[\sin(A + B) - \sin(A - B)] \
\cos A \cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)] \
\sin A \sin B = -\frac{1}{2}[\cos(A + B) - \cos(A - B)] \] These formulas are essential for transforming the product of two trigonometric functions into a sum or difference, which can then be integrated into complex math problem solving, such as in the provided IIT-JEE exercise.

Math Problem Solving

Math problem solving is a strategic process that involves understanding the problem, devising a plan, carrying out the plan, and reviewing the solution. It's a critical skill for all students, particularly for those preparing for competitive exams. In the context of trigonometry and IIT-JEE preparation, problem solving requires familiarity with trigonometric identities, formulas, and the ability to manipulate them effectively.

Approaching a Trigonometric Problem

First, understand what is given and what needs to be found. Then, identify which trigonometric identities or formulas can simplify the problem. Finally, execute step-by-step algebraic manipulations, keeping in mind the ultimate goal of finding the simplest form that reveals the solution. By practicing a variety of problems, students enhance their problem-solving skills and become better equipped to tackle the high-pressure environment of competitive exams like the IIT-JEE.

Trigonometry for Competitive Exams

Trigonometry is a significant component of the mathematics section in many competitive exams, including IIT-JEE. Being adept at trigonometry involves more than just memorizing formulas; it's about understanding how to apply them in various contexts and developing quickness in problem identification and resolution.

Tips for Mastering Trigonometry

• Practice regularly to become familiar with different types of trigonometric problems.
• Learn to recognize patterns and apply the appropriate formulas quickly.
• Time yourself while practicing to improve speed and accuracy.
• Pay special attention to common pitfalls and tricky questions that can appear in exams.

A sound strategy for trigonometry problems can make a substantial difference in your performance in competitive exams.