To solve the equation [tex]\(\cos(11b + 2) = \sin(12b - 4)\)[/tex], we can use a trigonometric identity. The identity that relates cosine and sine is given by:
[tex]\[
\cos(x) = \sin(90^\circ - x)
\][/tex]
Applying this identity, we can rewrite the original equation [tex]\(\cos(11b + 2)\)[/tex] as:
[tex]\[
\cos(11b + 2) = \sin(90^\circ - (11b + 2))
\][/tex]
So, the given equation [tex]\(\cos(11b + 2) = \sin(12b - 4)\)[/tex] can be rewritten using this identity as:
[tex]\[
\sin(90 - (11b + 2)) = \sin(12b - 4)
\][/tex]
Simplify the argument of the sine function on the left side:
[tex]\[
\sin(90 - 11b - 2) = \sin(12b - 4)
\][/tex]
[tex]\[
\sin(88 - 11b) = \sin(12b - 4)
\][/tex]
For the sine of two angles to be equal, the angles themselves must either be equal, or they must sum to [tex]\(180^\circ\)[/tex] given the periodic nature of the sine function. Therefore, we have:
[tex]\[
88 - 11b = 12b - 4
\][/tex]
or
[tex]\[
88 - 11b = 180^\circ - (12b - 4)
\][/tex]
Let’s solve the first equation:
[tex]\[
88 - 11b = 12b - 4
\][/tex]
Move all terms involving [tex]\(b\)[/tex] to one side and constants to the other:
[tex]\[
88 + 4 = 12b + 11b
\][/tex]
[tex]\[
92 = 23b
\][/tex]
Divide both sides by 23:
[tex]\[
b = \frac{92}{23}
\][/tex]
Simplify:
[tex]\[
b = 4
\][/tex]
Therefore, the value of [tex]\(b\)[/tex] is [tex]\( \boxed{4} \)[/tex].
