First slide

Theory of expressions

Question

The equation $e^{\sin x} - e^{- \sin x} = 4$ has:

Moderate

A

no real roots
B

exactly one real root
C

exactly four real roots
D

infinite number of real root

Solution

Put $e^{\sin x} = y .$ Note that $1 / e \leq y \leq e .$
Also, the given equation can be written as
$\begin{array}{l} y - 1 / y = 4 or y^{2} - 4 y - 1 = 0 \\ \Rightarrow y = 2 \pm \sqrt{5} . \end{array}$
As $1 / e \leq y \leq e,$ none of the two values of $y$ is possible.

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