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



Answer :

To solve for [tex]\(x\)[/tex] in the equation [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], where [tex]\(0^\circ < x < 90^\circ\)[/tex], we can use a trigonometric identity to transform the equation into a simpler form. The key identity we will use is:

[tex]\[ \sin(90^\circ - \theta) = \cos(\theta) \][/tex]

Given [tex]\(\cos(x) = \sin(14^\circ)\)[/tex], this identity suggests that:

[tex]\[ \cos(x) = \sin(90^\circ - x) \][/tex]

For our specific case, this means:

[tex]\[ \cos(x) = \sin(14^\circ) \][/tex]

By the identity [tex]\(\sin(90^\circ - \theta) = \cos(\theta)\)[/tex], we can equate:

[tex]\[ \sin(14^\circ) = \cos(76^\circ) \][/tex]

Thus,

[tex]\[ \cos(x) = \cos(76^\circ) \][/tex]

Since the cosine function is unique in the interval [tex]\(0^\circ < x < 90^\circ\)[/tex], comparing both sides, we have:

[tex]\[ x = 76^\circ \][/tex]

Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(\cos(x) = \sin(14^\circ)\)[/tex] is:

[tex]\[ \boxed{76^\circ} \][/tex]