MATH 221

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 30 July 2014

Lecture 7

POOF! ... and there etched in stone was eix=cosx+isinx. Do you believe it?

Verify cos(x+y) = cosxcosy-sinxsiny , sin(x+y) = sinxcosy+sinxcosy. Well eixeiy = (cosx+isinx) (cosy+isiny) = cosxcosy+icosx siny+isinx cosy+i2sinxsiny = cosxcosy- sinxsiny+i (sinxcosy+cosx+siny). On the other hand eixeiy= ei(x+y)= cos(x+y)+isin(x+y). So cos(x+y) = cosxcosy-sinxsiny and sin(x+y) = sinxcosy+sinxcosy.

Verify sinx = x-x33!+ x55!- x77!+ x99!-, cosx = 1- x22!+ x44!- x66!+ x88!-. Well eix = 1+ix+ (ix)22!+ (ix)33!+ (ix)44!+ (ix)55!+ = 1+ix+ i2x22!+ i3x33!+ i4x44!+ i5x55!+ i6x66!+ = 1 +ix -x22! -ix33! +x44! +ix55! -x66! -ix77! + 1 +ix -x22! -ix33! +x44! +ix55! -x66! -ix77! + = ( 1- x22!+ x44!- x66!+ ) +i ( x- x33!+ x55!- x77!+ ) . On the other hand eix=cosx+isinx. So cosx = ( 1- x22!+ x44!- x66!+ ) and sinx = ( x- x33!+ x55!- x77!+ ) .

Verify sin(-x)=-sinx and cos(-x)=cosx. sin(-x) = (-x)- (-x)33!+ (-x)55!- (-x)77!+ = -x- -x33!+ -x55!- -x77!+ = -x- -x33!+ -x55!- -x77!+ = - ( x- x33!+ x55!- x77!+ ) =-sinx. cos(-x) = 1- (-x)22!+ (-x)44!- (-x)66!+ (-x)88!- = 1- x22!+ x44!- x66!+ x88!- = cosx.

Verify sin2x+cos2x=1.

Well eixe-ix= eix-ix=e0=1. On the other hand eixe-ix = (cosx+isinx) (cos(-x)+isin(-x)) = (cosx+isinx) (cosx-isinx) = cos2x-icosx sinx+isinxcosx -i2sin2x = cos2x+sin2x. So 1=cos2x+sin2x

Find dydx when y=sinx. dydx = dsinxdx = d ( x- x33!+ x55!- x77!+ ) dx = 1- 3x23·2·1+ 5x45·4·3·2- 7x67·6·5·4·3·2·1+ = 1- x22·1+ x44·3·2- x66·5·4·3·2+ = 1- x22!+ x44!- x66!+ = cosx. So dsinxdx =cosx.

Find dcosxdx. dcosxdx = d ( 1- x22!+ x44!- x66!+ x88!- x1010!+ ) dx = -2x2·1 +4x34·3·2·1 -6x56·5·4·3·2 +8x78·7·6·5·4·3·2 - = -x +x33·2·1 -x55·4·3·2 +x77·6·5·4·3·2 - = -x +x33! -x55! +x77! - = - ( x- x33!+ x55!- x77!+ ) = -sinx. So dcosxdx =-sinx.

Inverse trig functions

The inverse function to f is the function that undoes f. sin-1x is the inverse function to sinx.
cos-1x is the inverse function to cosx.
tan-1x is the inverse function to tanx.
cot-1x is the inverse function to cotx.
sec-1x is the inverse function to secx.
csc-1x is the inverse function to cscx.
So sin-1(sinB) =Bandsin (sin-1x)=x. If y=sinx then sin-1y=x. Since 0=sin0 then sin-10=0, and 1=sinπ2 so sin-11=π2.

Warnings

(1) f(x)=sin-1x is not a function in the strictest sense. sin-10= 0or πor 2πor -πor -2πor 10πor This is similar to 9, which is 3 or -3 depending on the context.
(2) cos-1x (cosx)-1. cos-1x is the function that undoes cosx. (cosx)-1= 1cosx. For example: (cos0)-1 = 1cos0=11 =1,and cos-10 = π2or -π2or 3π2or -3π2or but cos-10 is never equal to 1.

Verify sin(cot-1x)=11+x2.

Let y=cot-1x. Then coty=x. 1+x2 x y 1 So sin(cot-1x) =siny=11+x2.

Verify sin-1(-x)=-sin-1x. sin-1(-x) =? -sin-1x sin(sin-1(-x)) =? sin(-sin-1x) -x =? -sin(sin-1x) -x =? -x YES!!

Verify tan-1x=cot-1(1x).

Say y=tan-1x. Then tany=x. So cot-1(1x)= cot-1(1tany)= cot-1(coty)=y =tan-1x.

Find dtanxdx. dtanxdx = d(sinxcosx)dx = d(sinx)(cosx)-1dx = sinxd(cosx)-1dx+ dsinxdx(cosx)-1 = sinx(-1) (cosx)-2 dcosxdx+ cosx1cosx = (-sinx)cos2x (-sinx)+1= sin2x+cos2x cos2x = 1cos2x = sec2x.

Find dtan-1xdx.

Set y=tan-1x. Then tany=x. Take the derivative: sec2ydydx=1. So dydx = 1sec2y= cos2y = (11+x2)2 =11+x2. 1+x2 1 y x

Find dsecxdx. dsecxdx = d1cosxdx= d(cosx)-1dx= -(cosx)-2 dcosxdx = -1cos2x (-sinx)= sinxcos2x= sinxcosx· 1cosx = tanxsecx.

Find dsec-1xdx.

Let y=sec-1x. Then secy=x. So tanysecy dydx=1. So dxdx= 1tanysecy. x 1 y 1+x2 So dydx= 1x2-1x. So dsec-1xdx= 1xx2-1.

Notes and References

These are a typed copy of Lecture 7 from a series of handwritten lecture notes for the class MATH 221 given on September 20, 2000.

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