HNC question signals

[Luke98]

hi there i was wondering if anyone could help me to work out and understand this question. its for my hnc course, i currently have no workings out for this question as im sturggling to find out what exactly i need to do.

The Question Reads;

"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?"

thanks,
luke

 

[Deleted member 4993]

hi there i was wondering if anyone could help me to work out and understand this question. its for my hnc course, i currently have no workings out for this question as im sturggling to find out what exactly i need to do.

The Question Reads;

"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?"

thanks,
luke

The problem above states:

V1=40sin(4t)​

V2=Acos(4t)​

​

The signal processor adds the signal together to form a third signal​

The first step would be to add those sine and cosine functions into one trig function (in this case sine function) - as given.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

 

[HallsofIvy]

Luke98 sent me a question by "conversation" which I thought should be posted here so others could see it and add their thoughts or learn from it:

"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?"

 

[HallsofIvy]

Use the trig identity sin(a+b)= sin(a)cos(b)+ cos(a)sin(b):
\(\displaystyle 50sin(4t+ \alpha)= 50sin(4t)cos(\alpha)+ 50cos(4t)sin(\alpha)\)

Further \(\displaystyle sin(2a)= 2 sin(a)cos(a)\) and \(\displaystyle cos(2a)= cos^2(a)- sin^2(a)\)
so \(\displaystyle sin(4t)= 2 sin(2t)cos(2t)= 2(2 sin(t)cos(t))(cos^2(t)- sin^2(t))= 4sin(t)cos^3(t)- 4sin^3(t)cos(t)\)
and \(\displaystyle cos(4t)= cos^2(2t)-sin^2(2t)= (cos^2(t)- sin^2(t))^2- (2sin(t)cos(t))^2= cos^4t)- 2sin^2(t)cos^2(t)+ sin^4- 4sin^2(t)cos^2(t)= cos^4(t)- 6cos^2(t)sin^2(t)+ sin^4(t)\).

 

[Luke98]

The problem above states:

V1=40sin(4t)​

V2=Acos(4t)​

​

The signal processor adds the signal together to form a third signal​

The first step would be to add those sine and cosine functions into one trig function (in this case sine function) - as given.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

The problem above states:

V1=40sin(4t)​

V2=Acos(4t)​

​

The signal processor adds the signal together to form a third signal​

The first step would be to add those sine and cosine functions into one trig function (in this case sine function) - as given.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520

Please share your work/thoughts about this assignment.

i just dont understand the question i see people using the sin(a+b) method i just dont know how to apply it with the 3 signals it has gave me to work out what A is ?

 

[Dr.Peterson]

hi there i was wondering if anyone could help me to work out and understand this question. its for my hnc course, i currently have no workings out for this question as im sturggling to find out what exactly i need to do.

The Question Reads;

"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?"

thanks,
luke

I asked you in your conversation with me to show some work when you submitted it. Surely there is something you can do. The way to learn to do something is to do something. You don't need to know exactly what to do.

Here is one place to start, since they showed you the form desired: expand [MATH]50\sin(4t + \alpha)[/MATH] using the identity [MATH]\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)[/MATH]. Then set this equal to [MATH]40\sin(4t) + A\cos(4t)[/MATH] and see what you find. Since the resulting equation has to be true for all t, you can equate the coefficients of [MATH]\sin(4t)[/MATH], and of [MATH]\cos(4t)[/MATH].

You don't need to do what Halls did; but show us something, so we can nudge you further.

 

[Luke98]

I asked you in your conversation with me to show some work when you submitted it. Surely there is something you can do. The way to learn to do something is to do something. You don't need to know exactly what to do.

Here is one place to start, since they showed you the form desired: expand [MATH]50\sin(4t + \alpha)[/MATH] using the identity [MATH]\sin(a+b) = \sin(a)\cos(b) + \cos(a)\sin(b)[/MATH]. Then set this equal to [MATH]40\sin(4t) + A\cos(4t)[/MATH] and see what you find. Since the resulting equation has to be true for all t, you can equate the coefficients of [MATH]\sin(4t)[/MATH], and of [MATH]\cos(4t)[/MATH].

You don't need to do what Halls did; but show us something, so we can nudge you further.

like this

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

 

[Dr.Peterson]

Yes, so what two pairs of expressions have to be equal in order for this to be an identity (true for all x)?

You'll be solving those two equations for A and a.

 

[Luke98]

Yes, so what two pairs of expressions have to be equal in order for this to be an identity (true for all x)?

You'll be solving those two equations for A and a.

im honestly not sure is there a more simple way you can put it without giving me the anwser.

 

[Deleted member 4993]

like this

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

50 * cos(a) * sin(4*t) + 50 * sin(a) * cos(4*t) = 40*sin(4*t) + A*cos(4*t)

Equating coefficients of sin(4*t) and cos(4*t) from the equation above, we can write:

50 * cos(a) = 40..........................................(1)

50 * sin(a) = A...........................................(2)

Continue,,,,,,

 

[Dr.Peterson]

You were told to solve for A, specifically, so focus on that. You need to find a way to combine these two equations to eliminate a.

What can you do to cos(a) and sin(a) that will result in a constant?

Even if you have no idea what to do, show us something, so that we can see you making progress. Even doing something wrong is progress, because we can help you learn from it what not to do, and point you in a slightly different direction.

 

[Luke98]

50 * cos(a) * sin(4*t) + 50 * sin(a) * cos(4*t) = 40*sin(4*t) + A*cos(4*t)

Equating coefficients of sin(4*t) and cos(4*t) from the equation above, we can write:

50 * cos(a) = 40..........................................(1)

50 * sin(a) = A...........................................(2)

Continue,,,,,,

for the following i got

50cos(a)=40 a=36.9
50sin(a)=A A=30

50cos(36.9)=40
50sin(36.9)=30

 

[Dr.Peterson]

So you found (correctly) that α = 36.9 DEGREES, and A = 30 (which turns out to be exact, if you use the Pythagorean identity I hinted at).

Plug those into V₂ = A cos(4t) and 50 sin(4t+α) and continue.

 

[Luke98]

so do i i just plot the graphs like so

f(x)=40sin(x)
f(x)=30cos(x)
f(x)=50sin+(x+36.9)

 

[Dr.Peterson]

"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 do i just plot the graphs like so

f(x)=40sin(x)
f(x)=30cos(x)
f(x)=50sin+(x+36.9)

I presume you meant

f(x)=40sin(4x)​

g(x)=30cos(4x)​

h(x)=50sin(4x+36.9)​

You may need to put α in radians, and graph in radians, which is what I expect for equations like this. This is probably what they mean by "plot/model the inputs and output".

 

[Lazur]

50 * cos(a) * sin(4*t) + 50 * sin(a) * cos(4*t) = 40*sin(4*t) + A*cos(4*t)

Equating coefficients of sin(4*t) and cos(4*t) from the equation above, we can write:

50 * cos(a) = 40..........................................(1)

50 * sin(a) = A...........................................(2)

Continue,,,,,,

Could you explain in steps how did you get to this point ? .

50 * cos(a) * sin(4*t) + 50 * sin(a) * cos(4*t) = 40*sin(4*t) + A*cos(4*t)

I done only as below and then got stuck and found this forum. Hope you be able to explain this to me .
From Compound Angle
sin(A+B) = sin(A)cos + cos(A)sin(B)
therefore
50sin(4t + α) = 50 [sin(4t)cos(α) + cos(4t)sin(α)]
and since 50sin(4t + α) = 40sin(4t) + Acos(4t)
therefore
50 [sin(4t)cos(α) + cos(4t)sin(α)] = 40sin(4t) + Acos(4t)

What do I do next ? Im lost like baby in the fog.

 

[Deleted member 4993]

Could you explain in steps how did you get to this point ? .

50 * cos(a) * sin(4*t) + 50 * sin(a) * cos(4*t) = 40*sin(4*t) + A*cos(4*t)

I done only as below and then got stuck and found this forum. Hope you be able to explain this to me .
From Compound Angle
sin(A+B) = sin(A)cos + cos(A)sin(B)
therefore
50sin(4t + α) = 50 [sin(4t)cos(α) + cos(4t)sin(α)]
and since 50sin(4t + α) = 40sin(4t) + Acos(4t)
therefore
50 [sin(4t)cos(α) + cos(4t)sin(α)] = 40sin(4t) + Acos(4t)

What do I do next ? Im lost like baby in the fog.

50 [sin(4t)cos(α) + cos(4t)sin(α)] = 40sin(4t) + Acos(4t)

First distribute 50:

50 [sin(4t)cos(α)] + 50 * [cos(4t)sin(α)] = 40sin(4t) + Acos(4t)​

Move everything to the Left-Hand-Side of the "=" sign:

{50 [sin(4t)cos(α)] - 40sin(4t)} + {50 [cos(4t)sin(α)] - Acos(4t)} = 0​

​

sin(4t) * {50 [cos(α)] - 40} + cos(4t) * {50 [sin(α)] - A} = 0​

Now look back at the solution again (#10) and try to figure out a path to get there