Simplify the equation - question about the angle

[Vital]

Homework Statement

Hello!
Please, take a look at the following equation and help me to understand where the authors got the value of φ = π/3. I don't see where it is derived from as no additional conditions are given.

Homework Equations

x(t) = 5e^(-t/5) cos(t) + 5e^(-t/5) √3 sin(t)

The Attempt at a Solution

Here is how I proceed:
x(t) = 5e^(-t/5) (cos(t) + √3 sin(t))

Now, given the formula:
f(x) = a sin(wx) + b cos(wx) + B (w > 0) is the same as
f(x) = √a²+b² sin(wx + φ) + B

In my case, B = 0, so I rewrite the expression as:
x(t) = 5e^(-t/5) ( √3 sin(t) + cos(t) )

x(t) = 5e^(-t/5) √√3²+1² sin(t + φ)

x(t) = 10e^(-t/5) sin(t + φ)

But in the book they have:
x(t) = 10e^(-t/5) sin(t + π/3)

Where did they get φ = π/3 from?

Thank you!

 

[Chestermiller]

x(t) = 5e^(-t/5) ( √3 sin(t) + cos(t) )=##10e^{-t/5}\left(\sin {t }\left(\frac{\sqrt{3}}{2}\right)+\cos{t}\left(\frac{1}{2}\right)\right)##

##\sin{(t+\phi)}=\sin t \cos{\phi}+\cos{t}\sin{\phi}##

What is the value of ##\phi##?

 

[Vital]

x(t) = 5e^(-t/5) ( √3 sin(t) + cos(t) )=##10e^{-t/5}\left(\sin {t }\left(\frac{\sqrt{3}}{2}\right)+\cos{t}\left(\frac{1}{2}\right)\right)##

##\sin{(t+\phi)}=\sin t \cos{\phi}+\cos{t}\sin{\phi}##

What is the value of ##\phi##?

Ah! I see. I should have computed the value of φ based on the coefficients' values:

coefficient1 x cos(φ) = √3
coefficient2 x sin(φ) = 1
(coefficient1 x cos(φ))² + (coefficient2 x sin(φ))² = coefficient²
hence, coefficient = 2, and thus cos(φ) = √3/2 and sin(φ) = ½, and of course φ = π/3
Now I see how the value appeared.

 

[Ray Vickson]

Ah! I see. I should have computed the value of φ based on the coefficients' values:

coefficient1 x cos(φ) = √3
coefficient2 x sin(φ) = 1
(coefficient1 x cos(φ))² + (coefficient2 x sin(φ))² = coefficient²
hence, coefficient = 2, and thus cos(φ) = √3/2 and sin(φ) = ½, and of course φ = π/3
Now I see how the value appeared.

You have been shown this more than once already in other threads. It is something you need to learn and commit to memory, especially if you are going into Phhsics or Electrical Engineering.

 

[Vital]

You have been shown this more than once already in other threads. It is something you need to learn and commit to memory, especially if you are going into Phhsics or Electrical Engineering.

It takes practice and time to remember these notions. I am doing my best, and as I am learning on my own I don't have enough practice. Happy to be able to ask here. :-)