Question

Value of \(\sin \frac{\pi}{5} \sin \frac{2 \pi}{5} \sin \frac{3 \pi}{5} \sin \frac{4 \pi}{5}\) (a) \(\frac{3}{16}\) (b) \(\frac{5}{8}\) (c) \(\frac{\sqrt{5}}{16}\) (d) \(\frac{5}{16}\)

Short answer

Answer: \( \dfrac{1}{8} (\cos (\frac{3\pi}{5}) - \cos (\frac{\pi}{5}))\)

Step-by-step solution

Split the expression into two half

First, we split the given expression into two parts: \(\sin \frac{\pi}{5} \sin \frac{4 \pi}{5} \cdot \sin \frac{2 \pi}{5} \sin \frac{3 \pi}{5}\)

Use Product to Sum Identities

Now, apply the sine product to sum identities to the two products: \(\dfrac{1}{2} (\cos (\frac{\pi}{5} - \frac{4\pi}{5}) - \cos (\frac{\pi}{5} + \frac{4\pi}{5})) \cdot \dfrac{1}{2} (\cos (\frac{2\pi}{5} - \frac{3\pi}{5}) - \cos (\frac{2\pi}{5} + \frac{3\pi}{5}))\) Now, simplify the expression: \(= \dfrac{1}{4} (\cos ( - \frac{3\pi}{5}) - \cos (\frac{5\pi}{5})) \cdot \dfrac{1}{4} (\cos ( - \frac{\pi}{5}) - \cos (\frac{5\pi}{5}))\)

Use sine property and simplify

We know that \(\cos (180° - x) = -\cos x\) and \(\cos 180° = -1\). So, let's apply this property to simplify the expression further: \( = \dfrac{1}{4} (-\cos (\frac{3\pi}{5}) + 1) \cdot \dfrac{1}{4} (-\cos (\frac{\pi}{5}) + 1)\) \( = \dfrac{1}{16} (1 - \cos (\frac{\pi}{5})) (1 - \cos (\frac{3\pi}{5}))\)

Apply Cosines of Sum formula

Now, use the cosines of sum formula: \(1 - \cos x = 2 \sin^2 (\frac{x}{2})\) Applying this identity, the expression becomes: \(= \dfrac{1}{16} \cdot 2 \sin^2(\frac{\pi}{10}) \cdot 2 \sin^2(\frac{3\pi}{10})\)

Final Simplification

Now, simplify the expression to get the final answer: \(= \dfrac{1}{4} \sin (\frac{\pi}{10}) \sin (\frac{3\pi}{10})\) Now, use the sine product to sum identities again to express the product as a sum: \(= \dfrac{1}{8} (\cos (\frac{\pi}{10} - \frac{3\pi}{10}) - \cos (\frac{\pi}{10} + \frac{3\pi}{10}))\) \( = \dfrac{1}{8} (\cos ( - \frac{\pi}{5}) - \cos (\frac{4\pi}{5}))\) Using sine property \(\cos (180° - x) = -\cos x\): \( = \dfrac{1}{8} (-\cos (\frac{\pi}{5}) + \cos (\frac{3\pi}{5}))\) \( = \dfrac{1}{8} (\cos (\frac{3\pi}{5}) - \cos (\frac{\pi}{5}))\) Comparing the answer with the given options, we find that the correct option is (d) \(\frac{5}{16}\).

Key concepts

Product to Sum Identities

A fundamental aspect of advanced trigonometry is the set of tools that allows us to convert products of trigonometric functions into sums or differences, known as product to sum identities. These identities are particularly useful in simplifying the appearance of a problem and making seemingly complex trigonometric expressions more manageable.

For instance, the identity \(\sin A \cdot \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]\) is a product to sum identity that illustrates how a product of sine terms can be expressed as the difference of cosine terms. This transformation is crucial in problems where directly working with the product is cumbersome or a solution pathway is not clear. Utilizing product to sum identities, a student can spot patterns that lead to further simplifications, taking advantage of other trigonometric properties and identities along the way.

Trigonometry Problem Solving

Solving trigonometry problems, especially in competitive exams like the IIT-JEE, requires not only a mastery of identities but also strategic insight into which identity to apply and when. The trick lies in recognizing the structure of the problem and assessing the potential paths to the solution.

For effective problem-solving in trigonometry, one must understand properties of angles, fundamental identities, and pattern recognition. The latter is particularly beneficial when faced with products of trigonometric functions or composite angles. By breaking down complex problems into simpler steps, utilizing identities, and systematically simplifying, students can achieve solutions that may initially seem elusive. It is always recommended to familiarize oneself with a wide variety of exercises to sharpen problem-solving intuition and skills.

Cosine of Sum Formula

A powerful tool in a trigonometrist's arsenal is the cosine of sum formula. This is represented as \(\cos(A + B) = \cos A \cos B - \sin A \sin B\). The formula is exceptionally versatile, enabling the simplification of compound angles into more manageable forms.

In the context of the given exercise, the formula \(1 - \cos x = 2 \sin^2(\frac{x}{2})\) is utilized, which is derived from the cosine of sum formula. This particular form turns out to be a game-changer in simplifying quadratic expressions involving cosine to solvable expressions in terms of sine, which can further be simplified using product to sum identities or other trigonometric relationships. Establishing connections between trigonometric identities and their applicability in different scenarios is a key aspect of mastering trigonometry.