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.
