Questions related to trig identities?

AKrishan17

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So basically i have this problem that states that c2*sin(wt)+c1*cos(wt)= A*Sin(wt+phi), where c2=Acos(Phi) and c1=Asin(Phi).
They ask me to find the amplitude of the function 2 sin(4pi*t)+5 cos(4pi*t), in terms of A sin(wt+phi). How do i do this?
 

Jomo

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Actually the helpers here get to ask how do you do this problem.
We are a help site which means we help students with their math problems.
Where do you need help? What have you tried? Please post post showing your work so we know where you need help.
If you had read our guidelines, you would have received help by now.
 

Dr.Peterson

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It happens that just yesterday I explained one way to solve this kind of question:


If you are taking a class, I would imagine you have been taught something about this; we'll want to know at least what you have learned that you can use.
 

Subhotosh Khan

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So basically i have this problem that states that

c2 * sin(wt) + c1 * cos(wt) = A * Sin(wt+phi), ............................................(1)​

where c2=Acos(Phi) and c1=Asin(Phi).​
They ask me to find the amplitude of the function 2 sin(4pi*t)+5 cos(4pi*t), in terms of A sin(wt+phi). How do i do this?
You are given a function:

2 * sin(4π*t) + 5 * cos(4π*t).................................................................(2)

referring to (1) - for function (2), what are:

c2 = ?

c1 = ?

\(\displaystyle \omega = ? \)
 

AKrishan17

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I am assuming that c2 is the 2 in front of the sine function and the c1 is the 5 in front of the cosine. The omega would be the 4pi?
 

Subhotosh Khan

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I am assuming that c2 is the 2 in front of the sine function and the c1 is the 5 in front of the cosine. The omega would be the 4pi?
Correct.

You were told:

c2 = A * cos(φ) = 2..................................(1)

and

c1 = A * sin(φ) = 5 .................................(2)

Calculate A from above two equations.

Hint:

cos2(φ) + sin2(φ) = 1
 

AKrishan17

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Somehow i got square root of 29 as A, is that correct?
 

Dr.Peterson

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Yes. Keep going. Now what is \(\displaystyle \varphi\)?
 

AKrishan17

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30 degrees? or Pi/6?
 

AKrishan17

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I don't really know what i did, but i'm stuck now.
 

Subhotosh Khan

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c2 = A * cos(φ) = 2..................................(1)

and

c1 = A * sin(φ) = 5 .................................(2)
What do you get when you divide (2) by (1)?
 

AKrishan17

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2.5
 

Subhotosh Khan

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I asked wrong question. Sorry about that. You have two equations:

A * cos(φ) = 2..................................(1)

and

A * sin(φ) = 5 .................................(2)

If you divide equation (2) by equation (1), you will get another equation:

What do you get on the Left-hand-side ?​
What do you get on the Right-hand-side ?​
 

AKrishan17

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is it equivalent to A tan phi = 5/2?
 

Subhotosh Khan

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is it equivalent to A tan phi = 5/2?
Almost - you get:

\(\displaystyle \frac{A}{A} * \frac{\sin(\phi)}{cos(\phi)} = \frac{5}{2}\)

we know A/A = 1, so:

\(\displaystyle \frac{\sin(\phi)}{cos(\phi)} = \frac{5}{2}\)

\(\displaystyle \tan(\phi) = 2.5\)

Now use your calculator and find out the value of \(\displaystyle \phi \).
 

AKrishan17

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I got 1.19? Is this correct?
 

AKrishan17

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radians haha!
 

Subhotosh Khan

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So basically i have this problem that states that c2*sin(wt)+c1*cos(wt)= A*Sin(wt+phi), where c2=Acos(Phi) and c1=Asin(Phi).
They ask me to find the amplitude of the function 2 sin(4pi*t)+5 cos(4pi*t), in terms of A sin(wt+phi). How do i do this?
So now - how would you answer your posted question?
 
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