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For what value of [tex]x[/tex] is [tex]\cos (x) = \sin (14^{\circ})[/tex], where [tex]0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}[/tex]?
A. [tex]28^{\circ}[/tex] B. [tex]76^{\circ}[/tex] C. [tex]31^{\circ}[/tex] D. [tex]14^{\circ}[/tex]
To find the value of [tex]\( x \)[/tex] for which [tex]\(\cos(x) = \sin(14^\circ) \)[/tex] within the range [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use a fundamental trigonometric identity.
The identity states that: [tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex] for any angle [tex]\(\theta\)[/tex].
Given [tex]\(\sin(14^\circ)\)[/tex], we need [tex]\(\cos(x)\)[/tex] to be equal to [tex]\(\sin(14^\circ)\)[/tex]. Using the identity, we can write: [tex]\[
\sin(14^\circ) = \cos(90^\circ - 14^\circ)
\][/tex]
Thus, if [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we must have: [tex]\[
x = 90^\circ - 14^\circ
\][/tex]
So, calculating this gives: [tex]\[
x = 90^\circ - 14^\circ = 76^\circ
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] for which [tex]\(\cos(x) = \sin(14^\circ) \)[/tex] is [tex]\( 76^\circ \)[/tex]. Hence, the correct answer is:
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