Amplitude of 3sin(x) + cos(x) ? (not 4...)?

Question

How do I find the amplitude. Please do not respond if you think it is 4. Also, please explain how you arrived at the answer.

Guys, I can do it with calculus. But there's a way using algebra. That's what I'm looking for. I believe the answer is 10^(1/2). but how do I get the answer.

Answer 1

Let:-
3 sin x + cos x = k. cos (x + a)
cos x + 3.sin x = (k.cos a).cos x - (k.sin a).sin x
1 = k cos a
3 = - k.sin a
1² + 3² = k² (cos²x + sin²x)
k² = 10
k = √10
Amplitude = √10

Answer 2

The easiest way would be by calculus, to take the derivative of y = 3 sin x + cos x and set it equal to 0. This would give you the values of x for which this function takes on a maximum or minimum. The amplitude would be the absolute function value at that point.
Without calculus, you could solve this graphically. That is, graph the equation and inspect where it takes the highest point. The amplitude would be this value.

Answer 3

Graph as many points as possible and you will see that the amplitude is reached between (Pi/2) and (3Pi/4) radians.

{EDIT} The poster below me has the correct answer and that technique can be used as long as the frequency of the 2 functions is identical. Since the frequency of both functions is 2Pi, the below technique can be successfully used. If the frequencies were different, calculus has to be utilized to determine the amplitude. Also, the first poster is incorrect because the value 4 is never reached due to the properties of the cosine function.

Answer 4

The amplitude of this expression would be 4 IF AND ONLY IF there was some value of x for which both would have a value of 1. This doesn't happen, so tough. The highest amplitude is somewhere above 3, which happens when 3cosx=sinx or Tan x =3.

Answer 5

Use the derivative to find the max and min. Subtract the min from the max then divide by 2 to get the amplitude

Answer 6

Hello,

Take the derivative and get

3cosx -sinx = 0 or |sinx/cosx| = 3 or |tanx| =3 or x = 71.5651 This is a max so 3sin(71.5651) + cos(71.5651) = 3.1623 Now
There is a min when the tan x = -3 at (180 + 71.5651) = 251.5651 )
3sin(251.5651) + cos(251.5651) = -3.1623 Now the amplitude is 3.1623 -(-3.1623) = 6.3246

Hope this helps.

Answer 7

You have to find the extrema & you do that by finding the deriviative & setting it equal to 0.
y=3 sin x + cos x
y'=0=3 cos x - sin x
3 cos x =sin x
3=sin x / cos x
tan x=3
x=arctan 3=71.565051177077989351572193720453

substitute this for x in
3 sin x + cos x=3.1622776601683793319988935444327

Answer 8

um pretty sure its just 3. look at your first set and take the coeffecient.