Calculus Volume 3

C | Review of Pre-Calculus

Calculus Volume 3C | Review of Pre-Calculus

Formulas from Geometry

$A=area,A=area,$ $V=Volume,andV=Volume,and$ $S=lateral surface areaS=lateral surface area$

Formulas from Algebra

Laws of Exponents

$xmxn=xm+nxmxn=xm−n(xm)n=xmn x−n=1xn(xy)n=xnyn(xy)n=xnyn x1/n=xnxyn=xnynxyn=xnyn xm/n=xmn=(xn)mxmxn=xm+nxmxn=xm−n(xm)n=xmn x−n=1xn(xy)n=xnyn(xy)n=xnyn x1/n=xnxyn=xnynxyn=xnyn xm/n=xmn=(xn)m$

Special Factorizations

$x2−y2=(x+y)(x−y)x3+y3=(x+y)(x2−xy+y2)x3−y3=(x−y)(x2+xy+y2)x2−y2=(x+y)(x−y)x3+y3=(x+y)(x2−xy+y2)x3−y3=(x−y)(x2+xy+y2)$

If $ax2+bx+c=0,ax2+bx+c=0,$ then $x=−b±b2−4ca2a.x=−b±b2−4ca2a.$

Binomial Theorem

$(a+b)n=an+(n1)an−1b+(n2)an−2b2+⋯+(nn−1)abn−1+bn,(a+b)n=an+(n1)an−1b+(n2)an−2b2+⋯+(nn−1)abn−1+bn,$

where $(nk)=n(n−1)(n−2)⋯(n−k+1)k(k−1)(k−2)⋯3⋅2⋅1=n!k!(n−k)!(nk)=n(n−1)(n−2)⋯(n−k+1)k(k−1)(k−2)⋯3⋅2⋅1=n!k!(n−k)!$

Formulas from Trigonometry

Right-Angle Trigonometry

$sinθ=opphypcscθ=hypoppcosθ=adjhypsecθ=hypadjtanθ=oppadjcotθ=adjoppsinθ=opphypcscθ=hypoppcosθ=adjhypsecθ=hypadjtanθ=oppadjcotθ=adjopp$

Trigonometric Functions of Important Angles

 $θθ$ $RadiansRadians$ $sinθsinθ$ $cosθcosθ$ $tanθtanθ$ $0°0°$ $00$ $00$ $11$ $00$ $30°30°$ $π/6π/6$ $1/21/2$ $3/23/2$ $3/33/3$ $45°45°$ $π/4π/4$ $2/22/2$ $2/22/2$ $11$ $60°60°$ $π/3π/3$ $3/23/2$ $1/21/2$ $33$ $90°90°$ $π/2π/2$ $11$ $00$ —

Fundamental Identities

$sin2θ+cos2θ=1sin(−θ)=−sinθ 1+tan2θ=sec2θcos(−θ)=cosθ1+cot2θ=csc2θtan(−θ)=−tanθsin(π2−θ)=cosθsin(θ+2π)=sinθ cos(π2−θ)=sinθcos(θ+2π)=cosθ tan(π2−θ)=cotθtan(θ+π)=tanθsin2θ+cos2θ=1sin(−θ)=−sinθ 1+tan2θ=sec2θcos(−θ)=cosθ1+cot2θ=csc2θtan(−θ)=−tanθsin(π2−θ)=cosθsin(θ+2π)=sinθ cos(π2−θ)=sinθcos(θ+2π)=cosθ tan(π2−θ)=cotθtan(θ+π)=tanθ$

Law of Sines

$sinAa=sinBb=sinCcsinAa=sinBb=sinCc$

Law of Cosines

$a2=b2+c2−2bccosAb2=a2+c2−2accosBc2=a2+b2−2abcosCa2=b2+c2−2bccosAb2=a2+c2−2accosBc2=a2+b2−2abcosC$

$sin(x+y)=sinxcosy+cosxsinysin(x−y)=sinxcosy−cosxsinycos(x+y)=cosxcosy−sinxsinycos(x−y)=cosxcosy+sinxsinytan(x+y)=tanx+tany1−tanxtanytan(x−y)=tanx−tany1+tanxtanysin(x+y)=sinxcosy+cosxsinysin(x−y)=sinxcosy−cosxsinycos(x+y)=cosxcosy−sinxsinycos(x−y)=cosxcosy+sinxsinytan(x+y)=tanx+tany1−tanxtanytan(x−y)=tanx−tany1+tanxtany$

Double-Angle Formulas

$sin2x=2sinxcosxcos2x=cos2x−sin2x=2cos2x−1=1−2sin2xtan2x=2tanx1−tan2xsin2x=2sinxcosxcos2x=cos2x−sin2x=2cos2x−1=1−2sin2xtan2x=2tanx1−tan2x$

Half-Angle Formulas

$sin2x=1−cos2x2cos2x=1+cos2x2sin2x=1−cos2x2cos2x=1+cos2x2$