Signals and Systems - Electrical Engineering

744 APPENDIX: Useful Formulas

Products

sin(θ)sin(φ)=

1

2

[cos(θ−φ)−cos(θ+φ)]

cos(θ)cos(φ)=

1

2

[cos(θ−φ)+cos(θ+φ)]

sin(θ)cos(φ)=

1

2

[sin(θ+φ)+sin(θ−φ)]

cos(θ)sin(φ)=

1

2

[sin(θ+φ)−sin(θ−φ)]

Euler’s Identity

ejθ=cos(θ)+jsin(θ) j=

√

− 1

cos(θ)=

ejθ+e−jθ

2

sin(θ)=

ejθ−e−jθ

2 j

tan(θ)=−j

[

ejθ−e−jθ

ejθ+e−jθ

]

Hyperbolic Trigonometry Relations

Hyperbolic cosine: cosh(α)=

1

2

(eα+e−α)

Hyperbolic sine: sinh(α)=

1

2

(eα−e−α)

cosh^2 (α)−sinh^2 (α)= 1

Calculus

Derivatives (u,vfunctions ofx;α,βconstants)

duv

dx

=u

dv

dx

+v

du

dx

dun

dx

=nun−^1

du

dx

Integrals

∫

φ(y)dx=

∫

φ(y)

y′

dy, wherey′=

dy

dx

∫

udv=uv−

∫

vdu

∫

xndx=

xn+^1

n+ 1

n6=−1, integer