Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Join our platform to connect with experts ready to provide precise answers to your questions in different areas.
Sagot :
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]
### 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]
We hope this information was helpful. Feel free to return anytime for more answers to your questions and concerns. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
