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Find the value of [tex]\( x \)[/tex] if [tex]\(\cos x = \sin 16^{\circ} \)[/tex].

A. [tex]\( 90^{\circ} \)[/tex]
B. [tex]\( 16^{\circ} \)[/tex]
C. [tex]\( 74^{\circ} \)[/tex]
D. cannot determine


Sagot :

To solve the problem, we need to determine the value of [tex]\( x \)[/tex] given that [tex]\(\cos x = \sin 16^\circ\)[/tex].

We start with the identity involving sine and cosine functions:

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

This means that for any angle [tex]\( \theta \)[/tex], the sine of [tex]\( \theta \)[/tex] is equal to the cosine of [tex]\( 90^\circ - \theta \)[/tex].

Given the problem:

[tex]\[ \cos x = \sin 16^\circ \][/tex]

Using the identity, we can write:

[tex]\[ \sin 16^\circ = \cos (90^\circ - 16^\circ) \][/tex]

Since [tex]\(\cos x = \sin 16^\circ\)[/tex], by comparing both sides, we have:

[tex]\[ \cos x = \cos (90^\circ - 16^\circ) \][/tex]

Thus:

[tex]\[ x = 90^\circ - 16^\circ \][/tex]

Simplifying the expression:

[tex]\[ x = 74^\circ \][/tex]

Therefore, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{74^\circ} \)[/tex].