# Thread: Minimum value of trigonometric expression?

1. ## Minimum value of trigonometric expression?

The minimum value of 27^cos x + 81^sin x is?

2. Does using AM-GM inequality help?

3. Hello,
Originally Posted by fardeen_gen
Does using AM-GM inequality help?
Yes !

$27=3^3$ and $81=3^4$

Therefore $27^{\cos(x)}+81^{\sin(x)}=3^{3 \cos(x)}+3^{4 \sin(x)}$

By AM-GM :
$3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$

So you just have to find the minimum of $3 \cos(x)+4 \sin(x)$

4. Originally Posted by fardeen_gen
The minimum value of 27^cos x + 81^sin x is?
This is periodic with perion $2 \pi$, so sketch the curve. This has a minimum near $x=4$. The minimum is a root of:

$f(x)=\ln(81) \cos(x) e^{\ln(81) \sin(x)}-\ln(27) \sin(x) e^{\ln(27) \cos(x)}$

Running the bisection method over this gives the minimum occurs at $x \approx 3.86147$, where the function is very close to $0.141494$ .

RonL

5. Originally Posted by Moo
Hello,

Yes !

$27=3^3$ and $81=3^4$

Therefore $27^{\cos(x)}+81^{\sin(x)}=3^{3 \cos(x)}+3^{4 \sin(x)}$

By AM-GM :
$3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$

So you just have to find the minimum of $3 \cos(x)+4 \sin(x)$
And how exactly do you show that the minimum of a lower bound is equal to the minimum of the function?

Note a numerical solution to the first problem is not equal to the function value at a minimum of $3 \cos(x)+4 \sin(x)$

RonL

6. @Moo: Minimum of 3 cos x + 4 sin x = - 5. But that means putting a negative quantity under root!

@CaptainBlack: The answer is 1/4. Doesn't match.

7. Originally Posted by CaptainBlack
And how exactly do you show that the minimum of a lower bound is equal to the minimum of the function?

Note a numerical solution to the first problem is not equal to the function value at a minimum of $3 \cos(x)+4 \sin(x)$

RonL
Then we can say that the minimum of the function will always be superior to the minimum of 3cos(x)+4sin(x)
But with the large inequality, I don't know, maybe it can be equal.

@Moo: Minimum of 3 cos x + 4 sin x = - 5.
No, because it's 3 to the power -5.

The answer is 1/4.
Gotta think about that then

8. Originally Posted by fardeen_gen
@Moo: Minimum of 3 cos x + 4 sin x = - 5. But that means putting a negative quantity under root!

@CaptainBlack: The answer is 1/4. Doesn't match.

Code:
>27^(cos(4))+81^(sin(4))
0.15193
>
There comes a point where numerical calculation over rules the answer at the back of the book.

RonL

9. If a cos x + b sin x = c, then - sqrt(a^2 + b^2) <= a cos x + b sin x <= + sqrt (a^2 + b^2). Therefore, for 3 cos x + 4 sin x, minimum seems to be -5.?? I am muddled up!

EDIT: My bad!!!! Its 3^(3 cos x + 4 sin x)

10. It's ok, I figured it out !

There is equality in AM GM if $3^{3 \cos(x)}=3^{4 \sin(x)}$

So let m be the minimum value of $2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$, that is $\frac 23 \sqrt{\frac 13}$ (substituting -5).

$3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m$

If $3^{3 \cos(x)}=3^{4 \sin(x)}$, then $3^{3 \cos(x)}+3^{4 \sin(x)}=2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m$

Thus m is the minimum, in particular when $3^{3 \cos(x)}=3^{4 \sin(x)}$

Sounds more correct ?

Edit : I have to work it out deeper, because I assumed that 3cos(x)=4sin(x)

11. Originally Posted by Moo
It's ok, I figured it out !

There is equality in AM GM if $3^{3 \cos(x)}=3^{4 \sin(x)}$

So let m be the minimum value of $2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$, that is $\frac 23 \sqrt{\frac 13}$ (substituting -5).

$3^{3 \cos(x)}+3^{4 \sin(x)} \ge 2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m$

If $3^{3 \cos(x)}=3^{4 \sin(x)}$, then $3^{3 \cos(x)}+3^{4 \sin(x)}=2 \sqrt{3^{3 \cos(x)+4 \sin(x)}} \ge m$

Thus m is the minimum, in particular when $3^{3 \cos(x)}=3^{4 \sin(x)}$

Sounds more correct ?

Edit : I have to work it out deeper, because I assumed that 3cos(x)=4sin(x)
Well it is true that the minimum of $2 \sqrt{3^{3 \cos(x)+4 \sin(x)}}$ is less than the minimum of $27^{\cos (x)} + 81^{\sin (x)}$, but the minimums do not occur at the same $x$, so what is this telling us?

RonL

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# minimum value of 27^cos x *81^sinx

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