The equation esinx – e–sin x

Engineering

Mathematics

Maximum and Minimum Value of Trigonometric Expression

Question

The equation e^sinx – e^{–sin x} – 4 = 0 has:

infinite number of real roots.

no real roots.

exactly one real root.

exactly four real root.

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Let esin x = k

$∴ k - \frac{1}{k} - 4 = 0 \Rightarrow k^{2} - 4 k - 1 = 0$

$k = e^{sinx} = 2 + \sqrt{5}$ = It is greater than e

⟹ Not possible and $e^{sinx} = 2 - \sqrt{5}$ = negative i.e. not possible.