# Minimize a certain function involving sine and cosine

• ShizukaSm
In summary, the conversation discusses a problem involving finding the minimum angle needed to move a box while doing the least amount of work possible. The original equation proposed was f(\theta)=\cos(\theta)+ 0.4sen(\theta), and after trying various methods, the equation was rewritten as f(\theta)=\sqrt{1-\sin^2(\theta)}+ 0.4sin(\theta). The conversation includes confusion about the usage of "sen" and "sin", as well as the method of finding the minimum angle. It is eventually determined that the point of the minimum is at \theta = 21.8°, and that the values of tan repeat every 180 degrees (pi in radians). The
ShizukaSm

## Homework Statement

It isn't a homework problem per se, but a curiosity a stumbled upon when trying to solve a physics problem (I was trying to calculate the angle I would need to do less work possible, while moving the box). The equation I found is:
$f(\theta)=\cos(\theta)+ 0.4sen(\theta)$

## Homework Equations

Just the one stated above, and trig identities, probably.

## The Attempt at a Solution

I tried a few things (including deriving and finding the roots of the function without success, since I couldn't found the roots), I also tried to rewrite (Using cos² + sin² = 1) and got:
$f(\theta)=\sqrt{1-\sin^2(\theta)}+ 0.4sin(\theta)$
But I also don't know how to find the minimum in this equation.

How could I go about solving that?

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ShizukaSm said:
I tried a few things (including deriving and finding the roots of the function without success, since I couldn't found the roots),

What problems did you run into? Did you do anything with tangent?

Interesting that you use both "sen" and "sin"! Can't decide between French and English?

You have $f(\theta)= cos(\theta)+ 0.4sin(\theta)$ (only English for me, I'm afraid.). I don't see any reason to introduce a square root just to have only sine. Taking the derivative, $f'(\theta)= -sin(\theta)+ 0.4cos(\theta)= 0$ at a max or min. That is the same as $sin(\theta)= 0.4 cos(\theta)$ or, since sine and cosine are not 0 for the same $\theta$, we must have $sin(\theta)/cos(\theta)= tan(\theta)= 0.4$. You can use a calculator to solve that.

Last edited by a moderator:
Reposting HallsofIvy's post to fix up the LaTex:
HallsofIvy said:
You have $f(\theta)= cos(\theta)+ 0.4sin(\theta)$. I don't see any reason to introduce a square root just to have only sine. Taking the derivative, $f'(\theta)= -sin(\theta)+ 0.4cos(\theta)= 0$ at a max or min. That is the same as $sin(\theta)= 0.4 cos(\theta)$ or, since sine and cosine are not 0 for the same $\theta$, we must have $sin(\theta)/cos(\theta)= tan(\theta)= 0.4$. You can use a calculator to solve that.

Oh god, I can't believe I ignored sin(x)/cos(x) = tan(x). I'm terribly sorry, thanks a lot! That solves my problem.

About the whole sen and sin thing, it's because I'm actually Brazilian, and here we write "Sen", so I sometimes get both of them confused :P

ShizukaSm said:
Oh god, I can't believe I ignored sin(x)/cos(x) = tan(x). I'm terribly sorry, thanks a lot! That solves my problem.

About the whole sen and sin thing, it's because I'm actually Brazilian, and here we write "Sen", so I sometimes get both of them confused :P

You also need to worry about whether the point you find is a maximizer or a minimizer of f.

Ray Vickson said:
You also need to worry about whether the point you find is a maximizer or a minimizer of f.

Yes indeed, but that was easily done by deriving again and finding if the result in this particular point would be positive or negative, since I found a negative value I concluded that it was a maximum, the value I wanted(In OP I miswrote, I actually wanted a maximum initially).

$tan(\theta)= 0.4;\theta = 21.8^o\\f''(21.8^o)= -1.077$

However, since you mentioned that, I just realized that I have no idea on how to find the minimum. Shouldn't $tan(\theta) = 0.4[\itex] yield two values, one of which is a maximum and another which is a minimum? ShizukaSm said: Yes indeed, but that was easily done by deriving again and finding if the result in this particular point would be positive or negative, since I found a negative value I concluded that it was a maximum, the value I wanted(In OP I miswrote, I actually wanted a maximum initially). [itex]tan(\theta)= 0.4;\theta = 21.8^o\\f''(21.8^o)= -1.077$

However, since you mentioned that, I just realized that I have no idea on how to find the minimum. Shouldn't [itex]tan(\theta) = 0.4[\itex] yield two values, one of which is a maximum and another which is a minimum?

Yes.

Ray Vickson said:
Yes.

Hmm. I found by trial and error that the angles should be: 21.8° and 201.8°, but how am I supposed to get the 201.8°? My calculator only gave me 21.8°.

ShizukaSm said:
Hmm. I found by trial and error that the angles should be: 21.8° and 201.8°, but how am I supposed to get the 201.8°? My calculator only gave me 21.8°.

tan(x)=tan(x+180). The values of tan repeat every 180 degrees (pi in radians). It's periodic with period pi.

Ohhh, I see, thanks!
I really need to get better in trigonometry. I have several wrong concepts :S

This is usually done with the identity

$$\cos (x)+\frac{2}{5} \sin (x)=\sqrt {1+\left( \frac{2}{5} \right) ^2 } \sin \left( x+\arctan \left( \frac{5}{2} \right) \right)$$

Which is quite easy to optimize.

## 1. How do you minimize a function involving sine and cosine?

To minimize a function involving sine and cosine, you can use the properties of these trigonometric functions, such as their periodicity and symmetry, to find the critical points. Then, use the first or second derivative test to determine if these points are local minima or maxima.

## 2. Can I use calculus to minimize a function with sine and cosine?

Yes, you can use calculus to minimize a function with sine and cosine. This involves finding the derivative of the function, setting it equal to zero, and solving for the critical points. Then, use the first or second derivative test to determine if these points are local minima or maxima.

## 3. Are there any special techniques for minimizing functions with sine and cosine?

One special technique for minimizing functions with sine and cosine is to use the double angle formula to rewrite the function in terms of only one trigonometric function. This can make it easier to find the critical points and determine the minimum value.

## 4. Can I use a graphing calculator to minimize a function with sine and cosine?

Yes, you can use a graphing calculator to minimize a function with sine and cosine. You can graph the function and use the trace or minimum feature to find the coordinates of the minimum point. However, it is important to also use calculus to verify that this point is indeed a minimum.

## 5. What applications use the minimization of functions with sine and cosine?

The minimization of functions with sine and cosine is commonly used in many fields of science and engineering, such as physics, chemistry, and economics. For example, it can be used to determine the optimal path for a projectile or to minimize the cost of a production process.

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