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For what value of [tex]x[/tex] is [tex]\sin(x) = \cos(32^{\circ})[/tex], where [tex]0^{\circ} \ \textless \ x \ \textless \ 90^{\circ}[/tex]?

A. [tex]64^{\circ}[/tex]
B. [tex]58^{\circ}[/tex]
C. [tex]32^{\circ}[/tex]
D. [tex]13^{\circ}[/tex]

Sagot :

To determine the value of [tex]\( x \)[/tex] such that [tex]\( \sin(x) = \cos(32^\circ) \)[/tex] for [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use the trigonometric identity that relates sine and cosine:

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

From the given problem, we have:

[tex]\[ \sin(x) = \cos(32^\circ) \][/tex]

Using the identity, we can rewrite [tex]\(\cos(32^\circ)\)[/tex]:

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

Thus, the equation becomes:

[tex]\[ \cos(90^\circ - x) = \cos(32^\circ) \][/tex]

For these two cosine values to be equal, their arguments must be equal (since cosine is a periodic function, and we are considering angles in the range [tex]\(0^\circ < x < 90^\circ\)[/tex] where it is uniquely one-to-one):

[tex]\[ 90^\circ - x = 32^\circ \][/tex]

To find [tex]\( x \)[/tex], we solve for [tex]\( x \)[/tex] in the equation:

[tex]\[ 90^\circ - x = 32^\circ \][/tex]

Subtract [tex]\( 32^\circ \)[/tex] from both sides:

[tex]\[ 90^\circ - 32^\circ = x \][/tex]

So:

[tex]\[ x = 58^\circ \][/tex]

Therefore, the value of [tex]\( x \)[/tex] that satisfies the equation [tex]\(\sin(x) = \cos(32^\circ)\)[/tex] is:

[tex]\[ \boxed{58^\circ} \][/tex]