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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]76^{\circ}[/tex]
B. [tex]14^{\circ}[/tex]
C. [tex]28^{\circ}[/tex]
D. [tex]31^{\circ}[/tex]


Sagot :

To find the value of [tex]\( x \)[/tex] such that [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] for [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a trigonometric identity:

We know that [tex]\(\sin(\theta) = \cos(90^\circ - \theta)\)[/tex].

Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], we can rewrite the right-hand side using the identity:
[tex]\[ \sin(14^\circ) = \cos(90^\circ - 14^\circ) \][/tex]

Now we have:
[tex]\[ \cos(x) = \cos(90^\circ - 14^\circ) \][/tex]

Since the cosine function is equal to itself if the angles are the same:
[tex]\[ x = 90^\circ - 14^\circ \][/tex]

Simplifying the expression:
[tex]\[ x = 76^\circ \][/tex]

So the value of [tex]\( x \)[/tex] is [tex]\(76^\circ\)[/tex].

Hence, the correct answer is:
A. [tex]\(76^{\circ}\)[/tex]