Answer:
Approximately 
.
Explanation:
![y_1 = 4.85\, \sin[(4.35\, x - 1270\, t) + 0]](https://tex.z-dn.net/?f=y_1%20%3D%204.85%5C%2C%20%5Csin%5B%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%20%2B%200%5D)
.
![y_2 = 4.85\, \sin[(4.35\, x - 1270\, t) + (-0.250)]](https://tex.z-dn.net/?f=y_2%20%3D%204.85%5C%2C%20%5Csin%5B%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%20%2B%20%28-0.250%29%5D)
.
Notice that sine waves 
and 
share the same frequency and wavelength. The only distinction between these two waves is the 
in 
.
Therefore, the sum 
would still be a sine wave. The amplitude of 
could be found without using calculus.
Consider the sum-of-angle identity for sine:
.
Compare the expression 
to . Let 
and 
. Apply the sum-of-angle identity of sine to rewrite .
![\begin{aligned}y_2 &= 4.85\, \sin[(\underbrace{4.35\, x - 1270\, t}_{a}) + (\underbrace{-0.250}_{b})]\\ &= 4.85 \, [\sin(4.35\, x - 1270\, t)\cdot \cos(-0.250) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7Dy_2%20%26%3D%204.85%5C%2C%20%5Csin%5B%28%5Cunderbrace%7B4.35%5C%2C%20x%20-%201270%5C%2C%20t%7D_%7Ba%7D%29%20%2B%20%28%5Cunderbrace%7B-0.250%7D_%7Bb%7D%29%5D%5C%5C%20%26%3D%204.85%20%5C%2C%20%5B%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Ccos%28-0.250%29%20%5C%5C%20%26%5Cquad%5Cquad%5Cquad%5C%3B%20%2B%20%5Ccos%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Csin%28-0.250%29%5D%20%5Cend%7Baligned%7D)
.
Therefore, the sum would become:
![\begin{aligned}& y_1 + y_2\\[0.5em] &= 4.85\, [\sin(4.35\, x - 1270\, t) \\ &\quad \quad \quad\;+\sin(4.35\, x - 1270\, t)\cdot \cos(-0.250) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \\[0.5em] &= 4.85\, [\sin(4.35\, x - 1270\, t)\cdot (1 + \cos(-0.250)) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%26%20y_1%20%2B%20y_2%5C%5C%5B0.5em%5D%20%26%3D%204.85%5C%2C%20%5B%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%20%5C%5C%20%26%5Cquad%20%5Cquad%20%5Cquad%5C%3B%2B%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Ccos%28-0.250%29%20%5C%5C%20%26%5Cquad%5Cquad%5Cquad%5C%3B%20%2B%20%5Ccos%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Csin%28-0.250%29%5D%20%5C%5C%5B0.5em%5D%20%26%3D%204.85%5C%2C%20%5B%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%281%20%2B%20%5Ccos%28-0.250%29%29%20%5C%5C%20%26%5Cquad%5Cquad%5Cquad%5C%3B%20%2B%20%5Ccos%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Csin%28-0.250%29%5D%20%5Cend%7Baligned%7D)
.
Consider: would it be possible to find 
and 
that satisfy the following hypothetical equation?
![\begin{aligned}& (4.85\, m)\cdot \sin((4.35\, x - 1270\, t) + c)\\&= 4.85\, [\sin(4.35\, x - 1270\, t)\cdot (1 + \cos(-0.250)) \\ &\quad\quad\quad\; + \cos(4.35\, x - 1270\, t)\cdot \sin(-0.250)] \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%26%20%284.85%5C%2C%20m%29%5Ccdot%20%5Csin%28%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%20%2B%20c%29%5C%5C%26%3D%204.85%5C%2C%20%5B%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%281%20%2B%20%5Ccos%28-0.250%29%29%20%5C%5C%20%26%5Cquad%5Cquad%5Cquad%5C%3B%20%2B%20%5Ccos%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Csin%28-0.250%29%5D%20%5Cend%7Baligned%7D)
.
Simplify this hypothetical equation:
.
Apply the sum-of-angle identity of sine to rewrite the left-hand side:
![\begin{aligned}& m\cdot \sin((4.35\, x - 1270\, t) + c)\\[0.5em]&=m\, \sin(4.35\, x - 1270\, t)\cdot \cos(c) \\ &\quad\quad + m\, \cos(4.35\, x - 1270\, t)\cdot \sin(c) \\[0.5em] &=\sin(4.35\, x - 1270\, t)\cdot (m\, \cos(c)) \\ &\quad\quad + \cos(4.35\, x - 1270\, t)\cdot (m\, \sin(c)) \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%26%20m%5Ccdot%20%5Csin%28%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%20%2B%20c%29%5C%5C%5B0.5em%5D%26%3Dm%5C%2C%20%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Ccos%28c%29%20%5C%5C%20%26%5Cquad%5Cquad%20%2B%20m%5C%2C%20%5Ccos%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%5Csin%28c%29%20%5C%5C%5B0.5em%5D%20%26%3D%5Csin%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%28m%5C%2C%20%5Ccos%28c%29%29%20%5C%5C%20%26%5Cquad%5Cquad%20%2B%20%5Ccos%284.35%5C%2C%20x%20-%201270%5C%2C%20t%29%5Ccdot%20%28m%5C%2C%20%5Csin%28c%29%29%20%5Cend%7Baligned%7D)
.
Compare this expression with the right-hand side. For this hypothetical equation to hold for all real 
and 
, the following should be satisfied:
, and
.
Consider the Pythagorean identity. For any real number 
:
.
Make use of the Pythagorean identity to solve this system of equations for . Square both sides of both equations:
.
.
Take the sum of these two equations.
Left-hand side:
.
Right-hand side:
.
Therefore:
.
.
Substitute 
back to the system to find . However, notice that the exact value of 
isn't required for finding the amplitude of 
.
(Side note: one possible value of is 
radians.)
As long as 
is a real number, the amplitude of would be equal to the absolute value of 
.
Therefore, the amplitude of would be:
.
