Maximizing (sin x+cos x) on [0,2*pi]

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In summary, the student was trying to find the maximum value of (sin x+cos x), which is in the first quadrant if x is in [0,2*pi]. If they knew the maximum possible value of (sin x+cos x), then the problem would be solved.
  • #1
ritwik06
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Homework Statement


I was solving a problem involving sine and cosine ratios.
If I come to know the maximum possible value of (sin x+cos x), x belongs to [0,2*pi]. my problm would be solved.

Homework Equations





The Attempt at a Solution


I think x ill be in 1st quadrant as both sin and cos are +ve. If I am not wrong x=45?
but how do i mathematically prove
this?
 
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  • #2
ritwik06 said:

Homework Statement


I was solving a problem involving sine and cosine ratios.
If I come to know the maximum possible value of (sin x+cos x), x belongs to [0,2*pi]. my problm would be solved.

Homework Equations





The Attempt at a Solution


I think x ill be in 1st quadrant as both sin and cos are +ve. If I am not wrong x=45?
but how do i mathematically prove
this?

One of the serious problems with showing no work at all is that we don't know what techniques you are familiar with. Do you know how to find the derivative of sin(x)+ cos(x)? Do you know what the derivative has to do with finding maximum values?
 
  • #3
HallsofIvy said:
One of the serious problems with showing no work at all is that we don't know what techniques you are familiar with. Do you know how to find the derivative of sin(x)+ cos(x)? Do you know what the derivative has to do with finding maximum values?

No I have no idea. Sorry. But I had already written what I could think of
 
  • #4
You may want to use the "R-formula", which states that [tex]a sin (x) \ + \ b cos(x) \ = \ R sin(x + \alpha)[/tex], where [tex] R \ = \ \sqrt{a^2+b^2} [/tex] and [tex] \alpha = tan^{-1} \frac{b}{a} [/tex].
 
  • #5
Since you haven't learned about derivatives, I'll nudge you in another direction that could help.
It uses trigonometric identities.

Hint: sin x + cos x = sin x + sin (90-x)
&
sin x + sin y = 2 sin ((x+y)/2)cos((x-y)/2)

This reduces the problem to finding the max of just one function instead of the max of a sum of functions.
 
Last edited:
  • #6
Alright, since the question was asked long ago, I shall go ahead and just give the answer, for my benefit as it passes the time (I've got to do SOMETHING at work, after all!)

[tex] \sin x + \cos x = \sin x + \sin (\pi/2 - x)[/tex]
[tex] \sin x + \sin (\pi/2 - x) = 2 \sin ((x + \pi/2 - x)/2)\cos((2x-\pi/2)/2)[/tex]
[tex]=2\cos(x-\pi/4)[/tex]
This must be maximized, but is easy to do. We know that [tex] \cos x [/tex] has a maximum of 1.
[tex] \cos (x-\pi/4) = 1 [/tex]
[tex] x-\pi/4 = \arccos 1 *[/tex]
[tex] x= \pi/4 [/tex]

*The only oversight here is not include that \arccos 1 = 2n\pi
 

What is the largest possible value in mathematics?

The largest possible value in mathematics depends on the type of number being considered. In the set of natural numbers, there is no largest value as the set is infinite. In the set of real numbers, the largest possible value is infinity, which represents a number that is greater than any real number.

What is the largest possible value in computer science?

In computer science, the largest possible value depends on the data type being used. For example, in a 32-bit integer, the largest possible value is 2,147,483,647. In a 64-bit integer, the largest possible value is 9,223,372,036,854,775,807. For floating-point numbers, the largest possible value is approximately 1.79 x 10^308.

What is the largest possible value in chemistry?

In chemistry, the largest possible value depends on the unit of measurement being used. For example, the largest possible value for mass would be the mass of the observable universe, which is approximately 10^53 kilograms. The largest possible value for temperature would be the Planck temperature, which is approximately 1.42 x 10^32 Kelvin.

What is the largest possible value in physics?

In physics, the largest possible value also depends on the unit of measurement being used. Some examples include the Planck length, which is approximately 1.6 x 10^-35 meters, and the Planck time, which is approximately 5.39 x 10^-44 seconds. Another commonly referenced large value in physics is the speed of light, which is approximately 3 x 10^8 meters per second.

Is there a limit to the largest possible value?

In mathematics and science, there is no limit to the largest possible value, as the concepts of infinity and infinitesimal allow for numbers to continue infinitely in both directions. However, in practical applications and calculations, there may be a limit due to the precision of measurement and the limitations of technology.

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