To find the value of [tex]\( x \)[/tex] such that [tex]\( \cos(x) = \sin(14^\circ) \)[/tex] within the range [tex]\( 0^\circ < x < 90^\circ \)[/tex], we can use a well-known trigonometric identity. Here's a step-by-step approach:
1. Understand the given equation:
We are given that [tex]\( \cos(x) = \sin(14^\circ) \)[/tex].
2. Recall the complementary angle identity:
The trigonometric identity states that [tex]\( \sin(90^\circ - \theta) = \cos(\theta) \)[/tex].
3. Apply the identity to the equation:
We can rewrite the right-hand side using this identity:
[tex]\[
\cos(x) = \sin(14^\circ) = \cos(76^\circ)
\][/tex]
This identity tells us that:
[tex]\[
\sin(14^\circ) = \cos(76^\circ)
\][/tex]
4. Equating the angles:
Since the cosine function [tex]\( \cos(x) \)[/tex] is equal to [tex]\( \cos(76^\circ) \)[/tex], it implies that:
[tex]\[
x = 76^\circ
\][/tex]
5. Verify the solution lies within the given range:
The value [tex]\( x = 76^\circ \)[/tex] indeed falls within the range [tex]\( 0^\circ < x < 90^\circ \)[/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]. This corresponds to Option B: 76 degrees.
