Compound Angle Formulae

[paulmc88]

Hi,

I am hoping someone can help me as my brain is fried tonight.

My HNC coursework is asking me to use Compound Angle Formula for the below question. Apologies if its been asked before I could not find a definitive solution in the threads.

The question below:


"Two signals are are sensed by a signal processor

V1=40sin(4t)
V2=Acos(4t)

The signal processor adds the signal together to form a third signal that can be described as;

50sin(4t+α)

Determine the value of A (the amplitude of ? 2 ).

Use to plot/model the inputs and output of the signal processor. How do you think graphical methods of sine wave combination compare with analytical methods?"

So this is where I am at

V1 + V2= Vo = 50sin(4t+a)

40sin(4t)+ Acos(4t)=50sin(4t+a)

sin(a+b)= (sinA)(cosB) + (cosA)(sinB)

40sin(4t)+ Acos(4t)= 50 sin(4t) cos(b) + cos(4t) sin(b)

40sin(4t)+ Acos(4t)= 50 cos(a) sin(4t) + 50 sin(a) cos(4t)

sin(4t):40 = 50cos(a)
cos(4t):A = 50sin(a)


Sin (4t) => 40 = 50 cos(a)
cos(a) = 40/ 50
a = cos-1 (40/50) = 0.64...(I think)... so sin(4t) = 50cos(0.64)

Then with the value of (a) I think I can replace (a) in the second part Cos(4t):A = 50sin(0.64) ..to find A...

However I am not sure how to figure A and if I was supposed to do 0.64 = sin-1(A/50) ?

And this is where I am so far trying to figure out how to find A from my original formula hoping that my 0.64 for (a) is correct.

Any help very much appreciated

Thanks

 

[tkhunny]

Okay,

Or "A" is found by the Pythagorean Theorem? [math]A^{2} + 40^{2} = 50^{2}[/math], A = 30 - With some consideration of sign and quadrant.

And the angle by the Arctangent? [math]\tan(\alpha) = \dfrac{30}{40}[/math]
And "sin(0.64)"? Where did the "4t+" go?

 

[Steven G]

Suppose 30 + 40 = x + y. Does this mean that x=30 and y=40? Yes, those results do satisfy the equation but there are many other solutions. Many pairs of numbers sum to 70.
Your logic is flawed-sorry. Try combining like terms and/or dividing by cos(4t).