Calculusi Notes

Pages: 3 (411 words) Published: June 23, 2013
CALCULUS (I) RULES

opp.
opp.
hyp.
hyp.
θ
θ
adj.
adj.
tanθ = sinθcosθ = opp.adj.
secθ = 1cosθ = hyp.adj.
cscθ = 1sinθ = hyp.opp.
cotθ = 1tanθ = cosθsinθ = adj.opp.

sin2θ + cos2θ = 1 sin2θ = 1 - cos2θ cos2θ = 1 - sin2θ tan2θ = sec2θ - 1 csc2θ = 1 + cot2θ sec2θ = 1 + tan2θ cot2θ = csc2θ - 1 S

S
A
A
- θ = 360 – θ
T
T
C
C
∴ sin(-θ) = - sinθ & csc(-θ) = - cosecθ
cos(-θ) = + cosθ & sec(-θ) = + secθ
tan(-θ) = - tanθ & cot(-θ) = - cotθ

sinπ2-x=cosx cosπ2-x=sinx tanπ2-x=cotx
cscπ2-x=secx secπ2-x=cscx cotπ2-x=tanx

sin(A±B) = sinAcosB ± cosAsinB
cos(A±B) = cosAcosB ∓ sinAsinB
tan(A±B)= tanA ± tanB1 ∓tanAtanB

∴ sin(2A) = 2sinAcosA
cos(2A) = cos2A – sin2A = 1 – 2sin2A = 2cos2A – 1
tan(2A) = 2tanA1- tan2A

sin2A = 12 - 12cos2A
cos2A = 12 + 12cos2A
tan2A = 1-cos⁡(2A)1+cos⁡(2A)

sinAsinB = 12 ( cos(A-B) – cos(A+B) )
cosAcosB = 12 ( cos(A-B) + cos(A+B) )
sinAcosB = 12 ( sin(A+B) + sin(A-B) )
cosAsinB = 12 ( sin(A+B) – sin(A-B) )

* ddxc=zero
* ddxcx=c
* ddxcy=cdydx
* ddxf(x)±g(x)=d(fx)dx±d(gx)dx
* ddxfx.g(x)=dfxdx.gx+dgxdx.f(x)
* ddxf(x)±g(x)=d(fx)dx±d(gx)dx
* ddxfx.gx.hx=
dfxdx.gx.hx+dgxdx.fx.hx+dhxdx.fx.g(x)
* ddxf(x)g(x)=dfxdx.gx-dgxdx.f(x)[gx]2
* dydx=dydu.dudv.dvdx , where y=fu, u=gv, v=h(x)
* ddxxn=n.xn-1
* ddxyn=n.yn-1.dydx ,where y=f(x)
* dydx=1dxdy

* ddxsinx=cosx
* ddxcscx=-cscx.cotx
* ddxcosx=-sinx
* ddxsecx=secx.tanx
* ddxtanx=sec2x
* ddxcotx=-csc2x

* xn dx=xn+1n+1+c, where n≠-1
* x-1 dx=lnx+c
* (ax+b)n dx= 1a(ax+b)n adx= 1a×(ax+b)n+1n+1+c
* sinx dx= -cosx+c
* cosx dx= sinx+c
* cosec2x dx= -cotx+c
* sec2x dx=...
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