Join the IDNLearn.com community and start getting the answers you need today. Our platform offers reliable and detailed answers, ensuring you have the information you need.
Sagot :
Certainly! Let's go through the process of finding the amplitude [tex]\( A \)[/tex] step-by-step for the expression [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex] relative to the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex].
### Step-by-Step Solution:
1. Given Expression and Desired Form:
- We start with the given expression [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex].
- We want to rewrite it in the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex].
2. Identify Coefficients:
- In [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex], we have:
[tex]\[ c_1 = 2 \quad \text{and} \quad c_2 = 5 \][/tex]
3. Use the Pythagorean Identity to Find Amplitude [tex]\( A \)[/tex]:
- According to the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex], the coefficients [tex]\( c_1 \)[/tex] and [tex]\( c_2 \)[/tex] can be expressed as:
[tex]\[ c_1 = A \sin \phi \quad \text{and} \quad c_2 = A \cos \phi \][/tex]
- Using the Pythagorean identity [tex]\((\sin^2 \phi + \cos^2 \phi = 1)\)[/tex], we can derive the amplitude [tex]\( A \)[/tex].
4. Calculate Amplitude [tex]\( A \)[/tex]:
- The amplitude [tex]\( A \)[/tex] is given by the formula:
[tex]\[ A = \sqrt{c_1^2 + c_2^2} \][/tex]
- Substitute the values of [tex]\( c_1 \)[/tex] and [tex]\( c_2 \)[/tex]:
[tex]\[ A = \sqrt{2^2 + 5^2} \][/tex]
- Simplify under the square root:
[tex]\[ A = \sqrt{4 + 25} \][/tex]
[tex]\[ A = \sqrt{29} \][/tex]
[tex]\[ A \approx 5.385164807134504 \][/tex]
### Conclusion
Thus, the amplitude [tex]\( A \)[/tex] for the expression [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex] when rewritten in the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex] is approximately [tex]\( 5.385164807134504 \)[/tex].
This step-by-step approach shows how the amplitude [tex]\( A \)[/tex] is derived using the coefficients of the sine and cosine terms along with the Pythagorean identity.
### Step-by-Step Solution:
1. Given Expression and Desired Form:
- We start with the given expression [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex].
- We want to rewrite it in the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex].
2. Identify Coefficients:
- In [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex], we have:
[tex]\[ c_1 = 2 \quad \text{and} \quad c_2 = 5 \][/tex]
3. Use the Pythagorean Identity to Find Amplitude [tex]\( A \)[/tex]:
- According to the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex], the coefficients [tex]\( c_1 \)[/tex] and [tex]\( c_2 \)[/tex] can be expressed as:
[tex]\[ c_1 = A \sin \phi \quad \text{and} \quad c_2 = A \cos \phi \][/tex]
- Using the Pythagorean identity [tex]\((\sin^2 \phi + \cos^2 \phi = 1)\)[/tex], we can derive the amplitude [tex]\( A \)[/tex].
4. Calculate Amplitude [tex]\( A \)[/tex]:
- The amplitude [tex]\( A \)[/tex] is given by the formula:
[tex]\[ A = \sqrt{c_1^2 + c_2^2} \][/tex]
- Substitute the values of [tex]\( c_1 \)[/tex] and [tex]\( c_2 \)[/tex]:
[tex]\[ A = \sqrt{2^2 + 5^2} \][/tex]
- Simplify under the square root:
[tex]\[ A = \sqrt{4 + 25} \][/tex]
[tex]\[ A = \sqrt{29} \][/tex]
[tex]\[ A \approx 5.385164807134504 \][/tex]
### Conclusion
Thus, the amplitude [tex]\( A \)[/tex] for the expression [tex]\( y(t) = 2 \sin 4\pi t + 5 \cos 4\pi t \)[/tex] when rewritten in the form [tex]\( y(t) = A \sin (\omega t + \phi) \)[/tex] is approximately [tex]\( 5.385164807134504 \)[/tex].
This step-by-step approach shows how the amplitude [tex]\( A \)[/tex] is derived using the coefficients of the sine and cosine terms along with the Pythagorean identity.
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Thank you for choosing IDNLearn.com for your queries. We’re here to provide accurate answers, so visit us again soon.
