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The value of sinpi/(14)sin(3pi)/(14)sin(...

The value of `sinpi/(14)sin(3pi)/(14)sin(5pi)/(14)sin(7pi)/(14)sin(9pi)/(14)sin(11pi)/(14)sin(13pi)/(14)` is equal to___________

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To solve the given expression \(\sin\left(\frac{\pi}{14}\right) \sin\left(\frac{3\pi}{14}\right) \sin\left(\frac{5\pi}{14}\right) \sin\left(\frac{7\pi}{14}\right) \sin\left(\frac{9\pi}{14}\right) \sin\left(\frac{11\pi}{14}\right) \sin\left(\frac{13\pi}{14}\right)\), follow these steps: ### Step 1: Use Symmetry Properties of Sine Function Recall that \(\sin(\pi - \theta) = \sin(\theta)\). This property helps us simplify the terms: - \(\sin\left(\frac{9\pi}{14}\right) = \sin\left(\pi - \frac{9\pi}{14}\right) = \sin\left(\frac{5\pi}{14}\right)\) - \(\sin\left(\frac{11\pi}{14}\right) = \sin\left(\pi - \frac{11\pi}{14}\right) = \sin\left(\frac{3\pi}{14}\right)\) - \(\sin\left(\frac{13\pi}{14}\right) = \sin\left(\pi - \frac{13\pi}{14}\right) = \sin\left(\frac{\pi}{14}\right)\) ...
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Knowledge Check

  • sin""(pi)/(14)sin""(3pi)/(14)sin""(5pi)/(14)sin""(7pi)/(14)sin""(9pi)/(14)sin""(11pi)/(14)sin""(13pi)/(14)=

    A
    `1//4`
    B
    `1//8`
    C
    `1//16`
    D
    `1//64`

  • The value of sin""(pi)/(14).sin""(3pi)/(14).sin""(5pi)/(14).sin""(7pi)/(14).sin""(9pi)/(14).sin""(11pi)/(14).sin""(13pi)/(14) is

    A
    a) `(1)/(8)`
    B
    b) `(1)/(16)`
    C
    c) `(1)/(32)`
    D
    d) `(1)/(64)`

  • The value of "sin"(pi)/(14)*"sin"(3pi)/(14)*"sin"(5pi)/(14) is

    A
    `1//4`
    B
    `1//8`
    C
    `1//2`
    D
    `1//16`

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