To find a possible solution to the equation [tex]\(\cos (x+2) = \sin (3x)\)[/tex], let's analyze the problem step-by-step to determine the correct value of [tex]\(x\)[/tex].
### Step 1: Understanding the Problem
We need to find an angle [tex]\(x\)[/tex] such that:
[tex]\[ \cos (x + 2^{\circ}) = \sin (3x) \][/tex]
### Step 2: Analyzing the Given Options
The possible values for [tex]\(x\)[/tex] provided are:
- [tex]\(0.5^{\circ}\)[/tex]
- [tex]\(1^{\circ}\)[/tex]
- [tex]\(22^{\circ}\)[/tex]
- [tex]\(44^{\circ}\)[/tex]
We'll check each of these values to see which one satisfies the equation.
### Step 3: Verify Each Option
#### Option A: [tex]\(x = 0.5^{\circ}\)[/tex]
- [tex]\(\cos (0.5^{\circ} + 2^{\circ}) = \cos (2.5^{\circ})\)[/tex]
- [tex]\(\sin (3 \times 0.5^{\circ}) = \sin (1.5^{\circ})\)[/tex]
Check if [tex]\(\cos (2.5^{\circ}) = \sin (1.5^{\circ})\)[/tex]:
[tex]\[\cos (2.5^{\circ}) \approx 0.999 \][/tex]
[tex]\[\sin (1.5^{\circ}) \approx 0.026\][/tex]
Since [tex]\(\cos (2.5^{\circ}) \neq \sin (1.5^{\circ})\)[/tex], this is not a correct solution.
#### Option B: [tex]\(x = 1^{\circ}\)[/tex]
- [tex]\(\cos (1^{\circ} + 2^{\circ}) = \cos (3^{\circ})\)[/tex]
- [tex]\(\sin (3 \times 1^{\circ}) = \sin (3^{\circ})\)[/tex]
Check if [tex]\(\cos (3^{\circ}) = \sin (3^{\circ})\)[/tex]:
[tex]\[\cos (3^{\circ}) \approx 0.998 \][/tex]
[tex]\[\sin (3^{\circ}) \approx 0.052\][/tex]
Since [tex]\(\cos (3^{\circ}) \neq \sin (3^{\circ})\)[/tex], this is not a correct solution.
#### Option C: [tex]\(x = 22^{\circ}\)[/tex]
- [tex]\(\cos (22^{\circ} + 2^{\circ}) = \cos (24^{\circ})\)[/tex]
- [tex]\(\sin (3 \times 22^{\circ}) = \sin (66^{\circ})\)[/tex]
Check if [tex]\(\cos (24^{\circ}) = \sin (66^{\circ})\)[/tex]:
[tex]\[\cos (24^{\circ}) \approx 0.913\][/tex]
[tex]\[\sin (66^{\circ}) \approx 0.913\][/tex]
Since [tex]\(\cos (24^{\circ}) = \sin (66^{\circ})\)[/tex], this satisfies the equation.
#### Option D: [tex]\(x = 44^{\circ}\)[/tex]
- [tex]\(\cos (44^{\circ} + 2^{\circ}) = \cos (46^{\circ})\)[/tex]
- [tex]\(\sin (3 \times 44^{\circ}) = \sin (132^{\circ})\)[/tex]
Check if [tex]\(\cos (46^{\circ}) = \sin (132^{\circ})\)[/tex]:
[tex]\[\cos (46^{\circ}) \approx 0.694\][/tex]
[tex]\[\sin (132^{\circ}) \approx 0.743\][/tex]
Since [tex]\(\cos (46^{\circ}) \neq \sin (132^{\circ})\)[/tex], this is not a correct solution.
### Conclusion
We have tested all supplied options, and the correct solution to the equation [tex]\(\cos (x+2) = \sin (3x)\)[/tex] is found to be:
[tex]\[ x = 22^{\circ} \][/tex]
Thus, the best answer is:
C. [tex]\(x = 22^{\circ}\)[/tex]
