To solve the equation [tex]\(3 \sin(2x) = 5 \cos(x)\)[/tex], let's go through the process step-by-step to rewrite it in the appropriate form.
1. Use the double-angle identity for sine:
[tex]\[\sin(2x) = 2 \sin(x) \cos(x)\][/tex]
2. Substitute this identity into the given equation:
[tex]\[3 \sin(2x) = 5 \cos(x)\][/tex]
[tex]\[3 \cdot 2 \sin(x) \cos(x) = 5 \cos(x)\][/tex]
3. Simplify the left-hand side of the equation:
[tex]\[6 \sin(x) \cos(x) = 5 \cos(x)\][/tex]
4. Factor out [tex]\(\cos(x)\)[/tex] from both sides of the equation:
[tex]\[\cos(x) \left(6 \sin(x) - 5\right) = 0\][/tex]
Therefore, the equation [tex]\(3 \sin(2x) = 5 \cos(x)\)[/tex] can be rewritten as:
[tex]\[ \cos(x) (6 \sin x - 5) = 0 \][/tex]
So, the correct answer is:
A. [tex]\(\cos x(6 \sin x - 5)=0\)[/tex]
