Question

The maximal value of f(x) = sin x - sqrt(3)*cos(x) (0º<=x<360º) is (?)

Answer 1

I used: sqrt(3) as tg 60º , then sin 60º/cos 60º

After a few steps, I got :

f(x) = sin(x-60)
----------
cos 60

Answer 2

It's: \[\sin x - \sqrt{3} * \cos x\]

Answer 3

\(\bf f(x)=sin(x)-\sqrt{3}cos(x)\) right?

Answer 4

I assume this is calculus, and thus you're meant to take the derivative of it, and find the maxima?

Answer 5

@jdoe0001 I tried to do that, but I got stuck after taking the derivative :(

Answer 6

alrite, hold the mayo for a few seconds

Answer 7

Difference rule and the product rule.

Answer 8

\(\bf f(x)=sin(x)-\sqrt{3}cos(x)
\\ \quad \\
\cfrac{dy}{dx}=cos(x)-[\sqrt{3}\cdot -sin(x)]\implies cos(x)+\sqrt{3}sin(x)
\\ \quad \\
\textit{setting the derivative to 0}
\\ \quad \\
0=cos(x)+\sqrt{3}sin(x)\implies -\sqrt{3}sin(x)=cos(x)
\\ \quad \\
\cfrac{-\sqrt{3}sin(x)}{cos(x)}=1\implies \cfrac{sin(x)}{cos(x)}=-\cfrac{1}{\sqrt{3}}\implies tan(x)=-\cfrac{1}{\sqrt{3}}
\\ \quad \\
x=tan^{-1}\left( -\cfrac{1}{\sqrt{3}} \right)\)

Answer 9

so... the tangent of it is negative, meaning either, the sine(y) or the cosine(x) is negative on those quadrants

and the only two quadrants that happens is II and IV :)

Answer 10

\[y = \sin x - \sqrt{3} \cos(x)\]

\[ = 2 \left( \frac{1}{2}\sin x - \frac{\sqrt{3}}{2} \cos x \right) \]

\[ = 2 \left( \sin \frac{\pi}{6} \sin x - \cos \frac{\pi}{6} \cos x \right) \]

\[ = - 2 \cos ( x + \frac{\pi}{6}) \]

see if that makes sense, could easily be a typo in there but that's an idea outside of calculus . then think about how a sinusioid oscillates :)

Answer 11

Where this come from ?\[\cfrac{-\sqrt{3}\sin(x)}{\cos(x)}=1\]

Answer 12

@IrishBoy123 hmm...

So this equation goes from -2 until 2, right?
I can see when x = 150º , -2*cos(180º) = -2 * (-1) = 2
Is this your idea?

2 is maximum, when x = 150º

Answer 13

I got:
\[\frac{ d }{ dx } =\sqrt{3}sinx+cosx\]

Answer 14

yes.

:)

Answer 15

I see. Thank you very much!

Answer 16

you can use calculus but you also get smart :-))

Answer 17

I think I can use the same thought in the expression I've found at the beginning:
f(x) = sin(x-60)
---------
cos 60

If we want the maximum, sin(of something here) must be 1. So 1/(1/2) will result in 2 as well.

Answer 18

hmmm, stilll wondering where \(\cfrac{-\sqrt{3}\sin(x)}{\cos(x)}=1\) comes from?

Answer 19

yes. that checks out. would never have seen it that way :)

Answer 20

@jdoe0001 yes... Is this a property?