To solve the equation [tex]\(\cos(50 + x)^\circ = \sin(2x - 6)^\circ\)[/tex], we can use trigonometric identities to simplify and solve for [tex]\(x\)[/tex].
First, recall the identity:
[tex]\[
\sin(\theta) = \cos(90^\circ - \theta)
\][/tex]
Using this identity, we can rewrite the right-hand side of the given equation:
[tex]\[
\sin(2x - 6)^\circ = \cos(90^\circ - (2x - 6)^\circ) = \cos(96^\circ - 2x)
\][/tex]
Now the equation becomes:
[tex]\[
\cos(50 + x)^\circ = \cos(96 - 2x)^\circ
\][/tex]
For [tex]\(\cos(A) = \cos(B)\)[/tex], we have two cases to consider:
1. [tex]\(A = B \)[/tex]
2. [tex]\(A = 360^\circ - B\)[/tex]
Applying these to our problem:
### Case 1: [tex]\(50 + x = 96 - 2x\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[
50 + x = 96 - 2x
\][/tex]
Combine like terms:
[tex]\[
50 + x + 2x = 96
\][/tex]
[tex]\[
50 + 3x = 96
\][/tex]
Subtract 50 from both sides:
[tex]\[
3x = 46
\][/tex]
Divide by 3:
[tex]\[
x = \frac{46}{3} \approx 15.3
\][/tex]
### Case 2: [tex]\(50 + x = 360^\circ - (96 - 2x)\)[/tex]
Simplify the right-hand side:
[tex]\[
50 + x = 360^\circ - 96 + 2x
\][/tex]
[tex]\[
50 + x = 264 + 2x
\][/tex]
Bring [tex]\(x\)[/tex] on one side:
[tex]\[
50 = 264 + x
\][/tex]
Subtract 264 from both sides:
[tex]\[
-214 = x
\][/tex]
Which is not a valid solution within the context.
Since only [tex]\(x = 15.3\)[/tex] makes sense, the nearest tenth value we get is:
[tex]\[
\boxed{15.3}
\][/tex]
Thus, the correct answer is [tex]\(15.3\)[/tex].
