Questions related to trig identities?

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

How to calculate maximum size of circles in rectangle?

Hi, For a given a rectangle with known width and height, I want to fit 4 circles of equal size regularly (see image) in such a way that the radius of the circles is maximized. The aspect ratio of the rectangle is such that 4 circles fit in the latticed way shown in the image. So the 'most...
www.freemathhelp.com

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.
 
AKrishan17 said:
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?
Click to expand...
You are given a function:

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

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

c2 = ?

c1 = ?

\(\displaystyle \omega = ? \)
 
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?
 
AKrishan17 said:
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?
Click to expand...
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 said:
2
Click to expand...
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 said:
is it equivalent to A tan phi = 5/2?
Click to expand...
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 said:
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?
Click to expand...
So now - how would you answer your posted question?
 
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