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hartnnBest ResponseYou've already chosen the best response.2
use sin Asin B formula, do u know it ?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
nopes, sin A  sin B on sin(x+h)  sin x.
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
I was thinking on starting out like this: \[\lim_{h \rightarrow 0}\frac{ \sin (x+h)  \sin(x) }{ h }\]
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
thats the correct start, i was giving a hint on how would u start simplifying numerator. using sin Asin B with A=x+h,B=x
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
us, would you expand sin(x+h)?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
i think that would also work, there u have to separate out denominator's h to all numerator terms.....i personally find it easier to do with sin Asin B...and less lengthy
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
how does sinAsinB work?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
sin A  sin B = 2 cos (A+B)/2 sin (AB)/ 2 right ? so sin (x+h)sin x = 2 cos (x+h/2) sin (h/2) ok ? now put h=0 directly in cos term and use sin t/t = 1 in sin term...
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
sorry, don't know sinAsinB = 2 cos((A+B)/2)sin((A+B)/2)
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
It's... actually \[sinA  sinB = 2cos\frac{A+B}{2}sin\frac{AB}{2}\]
 one year ago

SkaematikBest ResponseYou've already chosen the best response.0
Q: Is that the product to sums formula?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
then u better go with sin(x+h) = sin x cos h + cos x sin h ....
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
sumtoproduct indeed.
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
lim... [ sin(x)cos(h)+cos(x)sin(h)sin(x) ] / h lim... [ sin(x)(cos(h)1)+cos(x)sin(h) ] / h then what?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
can u solve lim x>0 (1cos x)/x = 1/2 ?? pretty standard. the other limit is pretty easy, just use sin h/h =1
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
Hmmm... Isn't it lim x>0 (1cos x)/x = 0?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
made a mistake....wait.yup its 0,sorry.
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
but basically u need to separate the numerator in 2 limits.
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
so your 1st limit will be 0 as pointed out correctly and 2nd limit will be just cos x *1 = cos x
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
\[\lim_{h \rightarrow 0}\frac{sin(x)(cos(h)1)+cos(x)sin(h) }{h}\]\[=\lim_{h \rightarrow 0}(\frac{sin(x)(cos(h)1)}{h}\ \ +\frac{cos(x)sin(h) }{h})\]\[=\lim_{h \rightarrow 0}sin(x)\frac{(cos(h)1)}{h} \ +\lim_{h \rightarrow 0}cos(x)\frac{sin(h) }{h}\]\[=...\]
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
Perhaps one more step for you... \[=sin(x)\lim_{h \rightarrow 0}\frac{(cos(h)1)}{h} \ +cos(x) \lim_{h \rightarrow 0}\frac{sin(h) }{h}\] I think you can evaluate the limit now..
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
how are theses limits calculated?
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
As @hartnn mentioned, use these: \[\lim_{x \rightarrow 0}\frac{1cos(x)}{x} =0\]\[\lim_{x \rightarrow 0}\frac{sin(x)}{x} =1\]
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
actually the 1st one isn't standard, u need to solve it by multiplying and dividing by (1+cos x) which gives sin^2 x in numerator and using 2nd limit to get answer as 0
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
yeah um, how are these proven? (Sorry if I'm being a bit fussy)
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
@apple_pi Do you have time to watch a video for explanations?
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
maybe later... just send me the links
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
there are many ways to prove lim sin x/x =1 the simplest one is the use of L'Hospitals rule, but since u are starting limits, i guess u don't know about this rule..
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
Oh yeah, MIT OCW is good. Does this show the whole d/dx sin(x)?
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
thanks...turns out I downloaded this ages ago, but stopped after lecture 2 and jumped about places
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
Here it goes. The notes I've copied from that lecture. IMG.pdf should contain all three pages.
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
Thanks a heap. Can you show me how sinAsinB = ...
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
Do you know how to work out the producttosum formula?
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
Example: sinAcosB = (1/2) [ sin(A+B) +sin(AB)]
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
that u can get using sin (xy) and sin (x+y) formula, u know that right ? just solve sin(x+y)  sin (xy) = ....... and then put x+y = A xy = B so x= (A+B)/2 y=(AB)/2.
 one year ago

CallistoBest ResponseYou've already chosen the best response.3
hartnn explained, I think :)
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
so whats sin(x+y)  sin (xy)= ?
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
sin(x)cos(y)+cos(x)sin(y)sin(x)cos(y)+cos(x)sin(y) = 2cos(x)sin(y)
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
thats correct, now just put values of x and y as i mentioned, u understood, how i found x and y in terms of A and b .. ?
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
x+y=A xy=B Adding x+x=A+B >x=(A+B)/2 i think y can can find now...
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
(A+B)/2+y=A y=(AB)/2
 one year ago

apple_piBest ResponseYou've already chosen the best response.0
uh, so does that mean sin(A)sin(B)=2cos((A+B)/2)sin((AB)/2)
 one year ago

hartnnBest ResponseYou've already chosen the best response.2
similarly u can find sin A+sin B cos A + cos B cos A  cos B and remember them as they are standard formulas.
 one year ago
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