The value of $\sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14}$ is equal to ..........

### Analyzing the Setup The objective is to evaluate the product: $$P = \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \sin \frac{7\pi}{14} \sin \frac{9\pi}{14} \sin \frac{11\pi}{14} \sin \frac{13\pi}{14}$$ The first step is to identify the anchor term. The middle term, $\sin \frac{7\pi}{14}$, simplifies to $\sin \frac{\pi}{2}$, which is exactly $1$. ### The Mirror Effect Observe the supplementary angle identity, $\sin(\pi - \theta) = \sin \theta$. Applying this to the remaining terms, we find: $\sin \frac{13\pi}{14} = \sin \frac{\pi}{14}$ $\sin \frac{11\pi}{14} = \sin \frac{3\pi}{14}$ $\sin \frac{9\pi}{14} = \sin \frac{5\pi}{14}$ Because of this symmetry, the product $P$ simplifies to the square of the first three terms: $$P = \left( \sin \frac{\pi}{14} \sin \frac{3\pi}{14} \sin \frac{5\pi}{14} \right)^2$$ ### The Transformation to Cosine To simplify the product, we use the complementary angle identity, $\sin \theta = \cos(\frac{\pi}{2} - \theta)$. Transforming the angles yields: $\sin \frac{\pi}{14} = \cos \frac{6\pi}{14} = \cos \frac{3\pi}{7}$ $\sin \frac{3\pi}{14} = \cos \frac{4\pi}{14} = \cos \frac{2\pi}{7}$ $\sin \frac{5\pi}{14} = \cos \frac{2\pi}{14} = \cos \frac{\pi}{7}$ Let $K$ be the product inside the square. We now have: $$K = \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{3\pi}{7}$$ ### The Geometric Progression We note that $\cos \frac{3\pi}{7} = \cos(\pi - \frac{4\pi}{7}) = -\cos \frac{4\pi}{7}$. Substituting this into $K$, we get: $$K = -\left( \cos \frac{\pi}{7} \cos \frac{2\pi}{7} \cos \frac{4\pi}{7} \right)$$ Using the identity $\prod_{k=0}^{n-1} \cos(2^k \theta) = \frac{\sin(2^n \theta)}{2^n \sin \theta}$ with $n=3$ and $\theta = \frac{\pi}{7}$: $$K = - \frac{\sin(8\pi/7)}{8 \sin(\pi/7)}$$ Since $\sin(8\pi/7) = \sin(\pi + \pi/7) = -\sin(\pi/7)$, the expression simplifies to: $$K = - \frac{-\sin(\pi/7)}{8 \sin(\pi/7)} = \frac{1}{8}$$ ### Final Calculation Finally, we compute the value of $P$: $$P = K^2 = \left( \frac{1}{8} \right)^2 = \mathbf{\frac{1}{64}}$$

I got stuck at $\cos \frac{3\pi}{7}$. How did you turn that into $-\cos \frac{4\pi}{7}$?

That is a brilliant observation! We used the supplementary angle identity for cosine: $\cos(\pi - \theta) = -\cos \theta$. So, $\cos \frac{3\pi}{7} = \cos(\pi - \frac{4\pi}{7}) = -\cos \frac{4\pi}{7}$. This step is crucial because it aligns the angles into the required $2^k$ geometric progression, allowing the formula to work perfectly.

JEE Advanced 1991

LEVELJEE Main

Animated Solution for Mathematics - Trigonometry: The value of $sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$ is equal to ..........

Visualized Solution

Analyze the Given Expression

Let the given expression be $P$ .
$P = sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$
Objective: Evaluate this product by identifying symmetry.

Evaluate the Middle Term

Identify the middle term: $sin \frac{7 π}{14}$
Simplify the angle: $\frac{7 π}{14} = \frac{π}{2}$
We know that $sin \frac{π}{2} = 1$ .
The expression reduces to six terms.

Apply Supplementary Angle Identity

Use the identity: $sin (π - θ) = sin θ$
Observe the last term: $sin \frac{13 π}{14} = sin (π - \frac{π}{14}) = sin \frac{π}{14}$
This matches the first term perfectly.

Reduce the Remaining Terms

Similarly, $sin \frac{11 π}{14} = sin (π - \frac{3 π}{14}) = sin \frac{3 π}{14}$
And $sin \frac{9 π}{14} = sin (π - \frac{5 π}{14}) = sin \frac{5 π}{14}$
The last three terms are identical to the first three terms.

Formulate the Squared Product

Substitute these symmetric values back into the product.
$P = (sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14}) \times (sin \frac{5 π}{14} sin \frac{3 π}{14} sin \frac{π}{14})$
Group them into a perfect square: $P = (sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14})^{2}$

Convert Sine to Cosine

Use the complementary angle identity: $sin θ = cos (\frac{π}{2} - θ)$
$sin \frac{π}{14} = cos (\frac{7 π}{14} - \frac{π}{14}) = cos \frac{6 π}{14} = cos \frac{3 π}{7}$
$sin \frac{3 π}{14} = cos (\frac{7 π}{14} - \frac{3 π}{14}) = cos \frac{4 π}{14} = cos \frac{2 π}{7}$
$sin \frac{5 π}{14} = cos (\frac{7 π}{14} - \frac{5 π}{14}) = cos \frac{2 π}{14} = cos \frac{π}{7}$

Rearrange into Geometric Progression

Let the term inside the square be $K$ .
$K = cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{3 π}{7}$
Notice that $cos \frac{3 π}{7} = cos (π - \frac{4 π}{7}) = - cos \frac{4 π}{7}$
Now, $K = - (cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{4 π}{7})$
The angles $\frac{π}{7}, \frac{2 π}{7}, \frac{4 π}{7}$ are in a geometric progression with a common ratio of $2$ .

Apply the Cosine Product Formula

Standard formula: $cos θ cos 2 θ cos 4 θ = \frac{s i n ( 2 ^{3} θ )}{2 ^{3} s i n θ}$
Substitute $θ = \frac{π}{7}$ :
$K = - \frac{s i n ( 8 \cdot \frac{π}{7} )}{8 s i n \frac{π}{7}} = - \frac{s i n \frac{8 π}{7}}{8 s i n \frac{π}{7}}$

Simplify the Numerator

Simplify the angle: $sin \frac{8 π}{7} = sin (π + \frac{π}{7})$
Using the third quadrant rule: $sin (π + \frac{π}{7}) = - sin \frac{π}{7}$
Substitute back: $K = - \frac{- s i n \frac{π}{7}}{8 s i n \frac{π}{7}} = \frac{1}{8}$

Final Calculation

Recall our original expression was $P = K^{2}$ .
Substitute the value of $K$ : $P = (\frac{1}{8})^{2}$
Final value: $P = \frac{1}{64}$
Key Takeaway: Always look for symmetry and geometric progressions in trigonometric products.

00:00 / 00:00

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The objective is to evaluate the product:

P = sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}

The first step is to identify the anchor term. The middle term,

sin \frac{7 π}{14}

, simplifies to

sin \frac{π}{2}

, which is exactly

1

The Mirror Effect

Observe the supplementary angle identity,

sin (π - θ) = sin θ

. Applying this to the remaining terms, we find:

sin \frac{13 π}{14} = sin \frac{π}{14}

sin \frac{11 π}{14} = sin \frac{3 π}{14}

sin \frac{9 π}{14} = sin \frac{5 π}{14}

Because of this symmetry, the product

P

simplifies to the square of the first three terms:

P = (sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14})^{2}

The Transformation to Cosine

To simplify the product, we use the complementary angle identity,

sin θ = cos (\frac{π}{2} - θ)

. Transforming the angles yields:

sin \frac{π}{14} = cos \frac{6 π}{14} = cos \frac{3 π}{7}

sin \frac{3 π}{14} = cos \frac{4 π}{14} = cos \frac{2 π}{7}

sin \frac{5 π}{14} = cos \frac{2 π}{14} = cos \frac{π}{7}

Let

K

be the product inside the square. We now have:

K = cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{3 π}{7}

The Geometric Progression

We note that

cos \frac{3 π}{7} = cos (π - \frac{4 π}{7}) = - cos \frac{4 π}{7}

. Substituting this into

K

, we get:

K = - (cos \frac{π}{7} cos \frac{2 π}{7} cos \frac{4 π}{7})

Using the identity

\prod_{k = 0}^{n - 1} cos (2^{k} θ) = \frac{s i n ( 2 ^{n} θ )}{2 ^{n} s i n θ}

with

n = 3

and

θ = \frac{π}{7}

K = - \frac{sin ( 8 π /7 )}{8 sin ( π /7 )}

Since

sin (8 π /7) = sin (π + π /7) = - sin (π /7)

, the expression simplifies to:

K = - \frac{- sin ( π /7 )}{8 sin ( π /7 )} = \frac{1}{8}

Final Calculation

Finally, we compute the value of

P

P = K^{2} = (\frac{1}{8})^{2} = \frac{1}{64}

Common Traps & FAQs

Why do we square the expression? Is that allowed?

We are not just squaring it arbitrarily! We identified that the terms in the product are symmetric. Specifically,

sin \frac{13 π}{14} = sin \frac{π}{14}

sin \frac{11 π}{14} = sin \frac{3 π}{14}

, and

sin \frac{9 π}{14} = sin \frac{5 π}{14}

. Since the middle term

sin \frac{7 π}{14} = 1

, the entire product

P

becomes

(sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14})^{2}

. We are simply grouping identical factors.

Why convert the sines to cosines? Can't we just solve the sine product directly?

You could, but it is much harder! The standard formula for the product of cosines with angles in geometric progression,

cos θ cos 2 θ cos 4 θ = \frac{s i n ( 8 θ )}{8 s i n θ}

, is a powerful tool in the JEE arsenal. Converting to cosine allows us to use this elegant identity, which collapses the entire product into a simple fraction.

I got stuck at

cos \frac{3 π}{7}

. How did you turn that into

- cos \frac{4 π}{7}

That is a brilliant observation! We used the supplementary angle identity for cosine:

cos (π - θ) = - cos θ

. So,

cos \frac{3 π}{7} = cos (π - \frac{4 π}{7}) = - cos \frac{4 π}{7}

. This step is crucial because it aligns the angles into the required

2^{k}

geometric progression, allowing the formula to work perfectly.

Animated Solution for Mathematics - Trigonometry: The value of $sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$ is equal to ..........

Visualized Solution

Analyze the Given Expression

Evaluate the Middle Term

Apply Supplementary Angle Identity

Reduce the Remaining Terms

Formulate the Squared Product

Convert Sine to Cosine

Rearrange into Geometric Progression

Apply the Cosine Product Formula

Simplify the Numerator

Final Calculation

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Mirror Effect

The Transformation to Cosine

The Geometric Progression

Final Calculation

Similar Questions

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Animated Solution for Mathematics - Trigonometry: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of sin14π​sin143π​sin145π​sin147π​sin149π​sin1411π​sin1413π​ is equal to ..........

Visualized Solution

Analyze the Given Expression

Evaluate the Middle Term

Apply Supplementary Angle Identity

Reduce the Remaining Terms

Formulate the Squared Product

Convert Sine to Cosine

Rearrange into Geometric Progression

Apply the Cosine Product Formula

Simplify the Numerator

Final Calculation

The Sigma Insight: Trigonometric Ratios and Identities

Analyzing the Setup

The Mirror Effect

The Transformation to Cosine

The Geometric Progression

Final Calculation

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Animated Solution for Mathematics - Trigonometry: The value of $sin \frac{π}{14} sin \frac{3 π}{14} sin \frac{5 π}{14} sin \frac{7 π}{14} sin \frac{9 π}{14} sin \frac{11 π}{14} sin \frac{13 π}{14}$ is equal to ..........

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