Answer :
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].
[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].
