Trig and hyperbolic expressions

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

Last updates: 17 June 2011

Trig and hyperbolic expressions

The exponential expression is ex = 1+x +x22! +x33! +x44! +x55! +x66! +x77! +.

The sine and cosine expressions are sinx = eix -e-ix 2i and cosx = eix +e-ix 2 , with i2=-1. The tangent, cotangent, secant and cosecant expressions are tanx = sinxcosx, cotx = 1tanx, secx = 1cosx, and cscx = 1sinx .

The hyperbolic sine and hyperbolic cosine expressions are sinhx= ex- e-x 2 and coshx= ex +e-x 2 . The hyperbolic tangent, hyperbolic cotangent, hyperbolic secant and hyperbolic cosecant functions are tanhx= sinhxcoshx, cothx = 1tanhx, sechx = 1coshx, and cschx= 1sinhx.

Example. Prove that eix =cosx+isinx .

Proof.
cosx+isinx = eix +e-ix 2 +i( eix -e-ix 2i) = eix +e-ix +eix -e-ix 2 = eix.

Example. Prove that sinx = x-x33! +x55! -x77! +x99! -x1111! +x1313! - and cosx = 1-x22! +x44! -x66! +x88! -x1010! +x1212! -.

Proof.
Compare coefficients of 1 and i on each side of cosx+isinx = eix = 1+ix+ (ix) 22! + (ix) 33! + (ix) 44! + (ix) 55! + (ix) 66! + (ix) 77! + = 1+ix +i2x2 2! + i3x3 3! + i4x4 4! + i5x5 5! + i6x6 6! + i7x7 7! + = 1+ix + i2x2 2! + i· i2x3 3! + (i2)2 x4 4! + i· (i2)2 x5 5! + (i2)3 x6 6! + i· (i2)3 x77! + = 1+ix + (-1) x2 2! + i·(-1) x3 3! + (-1)2 x44! + i· (-1)2 x55! + (-1)3 x6 6! + i· (-1)3 x7 7! + = 1+ix -x22! -ix33! +x44! +ix55! -x66! -ix77! + = ( 1- x22! +x44! -x66! +) +i( x- x33! +x55! -x77! +) .

Example. Prove that cos(-x) =cosx and sin(-x) = -sinx.

Proof.
cos(-x) = ei(-x) + e-i(-x) 2 = e-ix +eix 2 =cosx. and sin(-x) = ei(-x) -e-i(-x) 2i = e-ix -eix 2i = -sinx.

Example. Prove that cos2x +sin2x=1 .

Proof.
1 = e0 = e ix+ (-ix) = eix e-ix = eix ei(-x) = (cosx+isinx) (cos(-x) +isin(-x)) = (cosx+isinx) (cosx-isinx) = cos2x -isinxcosx +isinxcosx -i2sin2x = cos2x -(-1)sin2x = cos2x +sin2x.

Example. Prove that cosx+y=cosxcosy-sinxsiny,andsinx+y=sinxcosy+cosxsiny.

Proof.
Comparing the coefficients of 1 and i on each side of cos(x+y) +isin(x+y) = ei(x+y) = eix+iy = eix eiy = (cosx+isinx) (cosy+isiny) = cosxcosy +icosxsiny +isinxcosy +i2sinxsiny = cosxcosy +icosxsiny +isinxcosy -sinxsiny = (cosxcosy -sinxsiny) +i (cosxsiny +sinxcosy) . gives cos(x+y) = cosxcosy -sinxsiny, and sin(x+y) = sinxcosy+cosxsiny .

Example. Prove that ex =coshx+sinhx.

Proof.

Example. Prove that cosh(-x) =coshx and sinh(-x) =-sinhx.

Proof.

Example. Prove that sinhx=x+x33!+x55!+x77!+x99!+x1111!+x1313!+ and coshx=1+x22!+x44!+x66!+x88!+x1010!+x1212!+.

Proof.

Example. Prove that cosh2x -sinh2x =1.

Proof.

Example. Prove that cosh(x+y) = coshxcoshy +sinhxsinhy, and sinh(x+y) = sinhxcoshy +coshxsinhy.

Proof.

Notes and References

The student should learn to do these proofs at the same time that the trig expressions are introduced.

References

[Bou] N. Bourbaki, Algebra II, Chapters 4–7 Translated from the 1981 French edition by P. M. Cohn and J. Howie, Reprint of the 1990 English edition, Springer-Verlag, Berlin, 2003. viii+461 pp. ISBN: 3-540-00706-7. MR1994218

[Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford University Press, 1995. ISBN: 0-19-853489-2 MR1354144

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