## seidi.yamauti 3 years ago What is the maximun of f(x)=3cosx+2sinx ? Whithout using derivatives

1. lgbasallote

2. Traxter

@seidi.yamauti are you aware of what the graphs of cosx and sinx look like?

3. Traxter

@abayomi12 he said without using derivatives.

4. abayomi12

ok

5. abayomi12

I'd simplify this first: f(x) = 2 * sin(x) * cos(x) f(x) = sin(2x)

6. matricked

max value of acosx+bsinx is sqrt(a^2+b^2) and min is -sqrt(a^2+b^2) thus here max value is sqrt(3^2+2^2)

7. mukushla

$a\cos x+b\sin x=\sqrt{a^2+b^2} (\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x)$ using the fact that$(\frac{a}{\sqrt{a^2+b^2}})^2+(\frac{b}{\sqrt{a^2+b^2}})^2=1$ u can prove$\sqrt{a^2+b^2} (\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x)=\sqrt{a^2+b^2} \sin(x+\alpha)$or$\sqrt{a^2+b^2} (\frac{a}{\sqrt{a^2+b^2}}\cos x+\frac{b}{\sqrt{a^2+b^2}}\sin x)=\sqrt{a^2+b^2} \cos(x+\beta)$

8. shubhamsrg

cool.. B|

9. phi

in other words, use the identity sin(x+y) = sin(x)cos(y)+cos(x)sin(y)

10. abayomi12

maximum = sqrt(3^2 + 2^2) = sqrt(13)

अगर आप चाहें तो वोल्फ्रम का उपयोग कर सकते हैं.

बात समझ आई?

@mukushla आप तो मास्टर हो, मास्टर!

14. mukushla

man this is hindi !

15. sami-21

ye cid walay yahan kya kar rahay hain :P

16. CID-Inspector-DAYA

@sami-21 Kya chal raha hai?

@sami-21 chutti pe hain hum sab

humari police

19. sami-21

btw just to let you know guys irreverent answers are considered as SPAM .

lol

21. mboorstin

You can use a special trig identity: Acos(x) + Bsin(x) = Ccos(x-y), where C=\sqrt{A^2=B^2) and y = arctan(B/A). Hence $f(x)=3\cos x + 2\sin x = \sqrt{13}\cos\left(x-arctan\tfrac{2}{3}\right)\approx\sqrt{13}\cos\left(x-.588\right)$ But in any case, the maximum for cos is 1, so the maximum for f(x) is the square root of 13.

22. satellite73

also $$a\sin(x)+b\cos(x)=\sqrt{a^2+b^2}\sin(x+\theta)$$ for suitable $$\theta$$ as i recall you can get this from "addition angle" formula

23. mboorstin

@satellite73, how are you inserting equations without linebreaks? $signs don't seems to work here (as they would in LaTeX). 24. satellite73 use \( instead of \[ 25. mboorstin Thank you. 26. satellite73 you are right,$ does not work here turns out \( is an alternative in latex if you need to see any code, right click and you can see it. it is good method for copying and pasting as well, so you don't have to rewrite the latex every time