The equation of a transverse wave traveling in a string is given by y(x,t) = 10 cos (π/2)(0.0050x - 8.0t + 0.57), in which x and y are expressed in centimeters and t in seconds. Write down the equation of a wave which, when added to the given one, would produce standing waves on the string.

The problem is...I don't know what to do with that phase constant in the way. Like, I know that if the equation was just:

y(x,t) = 10 cos (π/2)(0.0050x - 8.0t)

I would just have to go through some really, really obnoxious trig to figure it out. But I am at a complete loss here. How do I even figure it out with that stupid phase constant around to muck things up?

add the two equations..

a) 10 cos (π/2)(0.0050x - 8.0t + 0.57)
b) 10 cos (π/2)(0.0050x - 8.0t+ PC) (your equation had PC at zero.

Now notice on both equations, the wavelength is the same, and the wave speed is the same. So lets make it simpler.
a) 10 cos (π/2)(A+.057)
b) 10 cos (π/2)(A+PC)

Remember the trig formula for sum of two cosines?cos u + cos v = 2 cos(½(u+v)) cos(½(u−v))

so the sum of the two waves is
20cos(1/2 (2A+PC+.057)cos(1/2(PC-.057))

The second term is a constant, which essentially van vary the amplitude of the prime first function. Notice what happens when PC-.057 is equal to Pi/2 or an odd multiple. In any event, the phase term only reduces the amplitude, event ot zero when PC is right. The first term gives standing waves.

But how does that even make sense?????? The equation for a standing wave is

y'(x,t) = 2Asin(kx)cos(omega * t)

And how do you go about finding the other phase constant?

To find the equation of a wave that, when added to the given wave, produces standing waves on the string, we need to adjust the equation so that the phase constant does not affect the resulting standing wave.

In the given equation:
y(x,t) = 10 cos [(π/2)(0.0050x - 8.0t + 0.57)]

The phase constant within the cosine function is 0.57. We can rewrite this equation in a different form to eliminate the phase constant's influence:

y(x,t) = 10 cos [(π/2)(0.0050x - 8.0t + φ)]

Here, φ represents the phase constant. To remove its effect, we want to choose a value for φ that eliminates the need for additional trigonometric calculations.

To find the appropriate value for φ, we can examine the behavior of standing waves. In a standing wave, the distance between crests or troughs is equal to a multiple of the wavelength λ.

For a stationary wave, we can use the following relationship:
0.0050x - 8.0t + φ = 2nπ

Where n is an arbitrary integer representing the number of half wavelengths.

To simplify the equation, we set φ = 2nπ - 0.0050x + 8.0t, where n is the desired number of nodes for the standing wave.

Therefore, the equation of a wave that, when added to the given wave, produces standing waves on the string is:

y(x, t) + y_s(x, t) = 10 cos [(π/2)(0.0050x - 8.0t + 2nπ - 0.0050x + 8.0t)]
y(x, t) + y_s(x, t) = 10 cos(nπ)

This equation results in standing waves with the desired number of nodes (n) for different integer values of n.