# Minimize a certain function involving sine and cosine

## 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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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?

HallsofIvy
Homework Helper
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.

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haruspex
Homework Helper
Gold Member
Reposting HallsofIvy's post to fix up the LaTex:
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

Ray Vickson
Homework Helper
Dearly Missed
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.

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 $tan(\theta) = 0.4[\itex] yield two values, one of which is a maximum and another which is a minimum? Ray Vickson Science Advisor Homework Helper Dearly Missed Yes indeed, but that was easily done by deriving again and finding if the result in this particular point would be positive or negative, since I found a negative value I concluded that it was a maximum, the value I wanted(In OP I miswrote, I actually wanted a maximum initially). [itex]tan(\theta)= 0.4;\theta = 21.8^o\\f''(21.8^o)= -1.077$

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.

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°.

Dick
Homework Helper
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

lurflurf
Homework Helper
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.