2 sin(π/22) sin(3π/22) sin(5π/22) sin(7π/22) sin(9π/22) is equal to (A) 3/16 (B) 1/16 (C) 1/32 (D) 9

Q: Why did we convert to cosine? Can't we just solve the sine product directly?

While theoretically possible, the product of sines doesn't have a clean, standard identity like the cosine product does. The cosine product identity $\prod_{k=1}^n \cos(\frac{k\pi}{2n+1}) = \frac{1}{2^n}$ is a powerful tool in your JEE arsenal. Converting to cosine is the strategic move that unlocks the problem.

Q: I got $1/32$ as my answer, but the options have $1/16$. Did I mess up?

You didn't mess up the math, you just missed the 'trap'! The original expression was $2P$, not $P$. You calculated $P$ correctly as $1/32$, but you must multiply by $2$ to get the final answer. Always re-read the question carefully!

Q: How do I know if $n=5$ is correct for the identity?

Look at the denominator of the angles in your cosine product. We have $\frac{k\pi}{11}$. The identity uses $\frac{k\pi}{2n+1}$. So, $2n+1 = 11$. Solving for $n$, we get $2n = 10$, so $n = 5$. Since our product goes from $k=1$ to $5$, it matches perfectly.

Question

$2 \sin \left(\frac{\pi}{22}\right) \sin \left(\frac{3 \pi}{22}\right) \sin \left(\frac{5 \pi}{22}\right) \sin \left(\frac{7 \pi}{22}\right) \sin \left(\frac{9 \pi}{22}\right)$ is equal to

Options:

(A) $\frac{3}{16}$

(B) $\frac{1}{16}$

(C) $\frac{1}{32}$

(D) $\frac{9}{32}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

The expression we are evaluating is $2 \sin(\frac{\pi}{22}) \sin(\frac{3\pi}{22}) \sin(\frac{5\pi}{22}) \sin(\frac{7\pi}{22}) \sin(\frac{9\pi}{22})$. It is natural to feel a sense of unease when looking at this jumble of trigonometric functions.

However, in the world of JEE Advanced, chaos is often just order in disguise. Let us define $P$ as the product of these five sine terms, such that our target expression is $2P$.

The angles involved are $\frac{\pi}{22}, \frac{3\pi}{22}, \frac{5\pi}{22}, \frac{7\pi}{22}, \frac{9\pi}{22}$. Notice that they are all odd multiples of $\frac{\pi}{22}$, which serves as our first clue.

### The Bridge to Cosine

The cosine function possesses an elegant identity for products of the form $\prod_{k=1}^{n} \cos(\frac{k\pi}{2n+1})$. To utilize this, we apply the complementary angle identity: $\sin(	heta) = \cos(\frac{\pi}{2} - 	heta)$.

We rewrite $\frac{\pi}{2}$ as $\frac{11\pi}{22}$. For any angle $\frac{k\pi}{22}$ in our product, the sine term transforms as follows:

$$\sin\left(\frac{k\pi}{22}ight) = \cos\left(\frac{11\pi}{22} - \frac{k\pi}{22}ight) = \cos\left(\frac{(11-k)\pi}{22}ight)$$

### The Transformation

Applying this transformation to each term in our product $P$:

For $k=1$, we get $\cos(\frac{10\pi}{22}) = \cos(\frac{5\pi}{11})$.
For $k=3$, we get $\cos(\frac{8\pi}{22}) = \cos(\frac{4\pi}{11})$.
For $k=5$, we get $\cos(\frac{6\pi}{22}) = \cos(\frac{3\pi}{11})$.
For $k=7$, we get $\cos(\frac{4\pi}{22}) = \cos(\frac{2\pi}{11})$.
For $k=9$, we get $\cos(\frac{2\pi}{22}) = \cos(\frac{\pi}{11})$.

Rearranging these in increasing order, our product $P$ becomes:

$$P = \prod_{k=1}^{5} \cos\left(\frac{k\pi}{11}ight)$$

### The Magical Identity

We utilize the standard trigonometric identity:

$$\prod_{k=1}^{n} \cos\left(\frac{k\pi}{2n+1}ight) = \frac{1}{2^n}$$

Setting $2n+1 = 11$, we find $2n = 10$, which implies $n = 5$. Since our product matches this form perfectly for $n=5$, we have:

$$P = \frac{1}{2^5} = \frac{1}{32}$$

### The Final Calculation

The most common mistake in JEE is to stop at the value of $P$. The original question asks for the value of $2P$.

We must multiply our result by $2$:

$$2 	imes P = 2 	imes \frac{1}{32} = \frac{1}{16}$$

The final value of the expression is **$\frac{1}{16}$**.

Animated Solution for Mathematics - Trigonometry: $2 sin (\frac{π}{22}) sin (\frac{3 π}{22}) sin (\frac{5 π}{22}) sin (\frac{7 π}{22}) sin (\frac{9 π}{22})$ is equal to

Select Answer:

Visualized Solution

Defining the Expression $P$

Using Complementary Angle Identity

Transforming Angles

Converting Sine to Cosine

Simplifying the Fractions

Rearranging in Increasing Order

The Cosine Product Identity

Matching Parameters

Calculating the Value of $P$

Final Calculation and Result

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Bridge to Cosine

The Transformation

The Magical Identity

The Final Calculation

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