To solve the given trigonometric equation [tex]$\sin (4x - 10)^{\circ} = \cos (40 - x)^{\circ}$[/tex], we need to use a trigonometric identity. One known identity is that [tex]\(\sin(A) = \cos(B)\)[/tex] if and only if [tex]\(A + B = 90^\circ\)[/tex].
Given the equation:
[tex]$ \sin (4x - 10)^{\circ} = \cos (40 - x)^{\circ} $[/tex]
We can use the identity [tex]\(\sin(A) = \cos(B)\)[/tex] implies:
[tex]$ (4x - 10) + (40 - x) = 90 $[/tex]
Now let's solve the equation step-by-step.
1. Combine the terms inside the parentheses:
[tex]$ 4x - 10 + 40 - x = 90 $[/tex]
2. Simplify the left-hand side:
[tex]$ 3x + 30 = 90 $[/tex]
3. Subtract 30 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]$ 3x = 60 $[/tex]
4. Divide both sides by 3 to solve for [tex]\(x\)[/tex]:
[tex]$ x = 20 $[/tex]
Therefore, the value of [tex]\(x\)[/tex] is:
[tex]$ \boxed{20} $[/tex]
So the correct answer is:
C. [tex]\(x = 20\)[/tex]
