The sum of angles formula for sine is , so you want and . Hence, .
When the two voltages V_{1} = 30sin(120Πt) and V_{2} = 40cos(120Πt)are applied to the same circuit, the resulting voltage V will be equal to their sum.
a. Graph the sum in the window [0 .05] by [-60,60] and use the graph to estimate values for a and Φ so that V = a sin(120Πt + Φ)
b. Use identities to verify that your expression for V is valid.
Okay........I got the graph on my calculator and can easily find a, it's just the amplitude of the sinusoid, which is 50.
So a=50, but how can I estimate a value for Φ from the graph? The answer in the back of the book is Φ = -5.353, but I can't see how to get there.
And for part b, how can I use identities to verify that 30sin(120Πt)+ 40cos(120Πt) = 50 sin(120Πt + (-5.353))????
Are you using degrees or radians? Angles are periodic (either 360 degrees or radians). Hence, there are an infinite number of answers. Either subtract 360 degrees or radians to see if your answer matches the book's.
To find phi, you would solve for it. You have
So, solving for gives
That should give a step function (it will have an infinite number of answers). So, pick any answer that satisfies the equations. I don't understand what you mean "how do you know to do that". Sine and cosine are always periodic. The book's answer is correct, but there are infinite correct answers. Your answer (before subtracting 2 pi) is correct. There is no need to subtract anything.
Edit: in the given range of , the graph gives the book's answer as the solution.
Right, looking a the graph must somehow make -5.353 stand out, but I cannot see how to interpret this value from the graph. I think it must be a phase shift of the graph of
50 sin(120Πt), but I can't see how to interpret this. Make sense? I'm just trying to understand how looking at the graph will give that value rather than using identities.
Here is a table of possible values for in the given interval: Table[Arcsin[(3/5)*Sin[120*Pi*t]+(4/5)*Cos[120*Pi*t]]-120*Pi*t,{t,0,0.05,0.001}] - Wolfram|Alpha