For what value of [tex]$x$[/tex] is [tex]$\cos (x)=\sin \left(14^{\circ}\right)$[/tex], where [tex][tex]$0^{\circ}\ \textless \ x\ \textless \ 90^{\circ}$[/tex][/tex]?

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



Answer :

To find the value of [tex]\( x \)[/tex] for which [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] given that [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use a property of complementary angles in trigonometry.

Here are the steps to solve the problem:

1. Recall the Complementary Angle Relationship: In trigonometry, the sine of an angle is equal to the cosine of its complementary angle. Specifically, for any angle [tex]\( \theta \)[/tex],
[tex]\[ \sin(90^\circ - \theta) = \cos(\theta) \][/tex]

2. Given Equation: We are given that
[tex]\[ \cos(x) = \sin(14^\circ) \][/tex]

3. Apply the Complementary Angle Property: According to the complementary angle property, we have
[tex]\[ \sin(14^\circ) = \cos(76^\circ) \][/tex]

4. Comparison: This tells us that
[tex]\[ \cos(x) = \cos(76^\circ) \][/tex]

5. Determine [tex]\( x \)[/tex]: Since cosine is a one-to-one function in the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can conclude that
[tex]\[ x = 76^\circ \][/tex]

Thus, the value of [tex]\( x \)[/tex] that satisfies [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] in the interval [tex]\( 0^\circ < x < 90^\circ \)[/tex] is [tex]\( 76^\circ \)[/tex].

So, the answer is [tex]\( \boxed{76^\circ} \)[/tex].