$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

Appendix

# Graphs of the Parent Functions

<figure id="CNX_Precalc_Figure_APP_001"></figure> <figure id="CNX_Precalc_Figure_APP_002"></figure> <figure id="CNX_Precalc_Figure_APP_003"></figure>

# Graphs of the Trigonometric Functions

<figure id="CNX_Precalc_Figure_APP_004"></figure> <figure id="CNX_Precalc_Figure_APP_005"></figure> <figure id="CNX_Precalc_Figure_APP_006"></figure> <figure id="CNX_Precalc_Figure_APP_007"></figure>

# Trigonometric Identities

 Pythagorean Identities cos[/itex] 2 t+ sin 2 t=1 1+ tan 2 t= sec 2 t 1+ cot 2 t= csc 2 t Even-Odd Identities cos(−t)=cost[/itex] sec(−t)=sec t sin(−t)=−sin t tan(−t)=−tan t csc(−t)=−csc t cot(−t)=−cot t Cofunction Identities cost=sin([/itex] π 2 −t ) sin t=cos( π 2 −t ) tan t=cot( π 2 −t ) cot t=tan( π 2 −t ) sec t=csc( π 2 −t ) csc t=sec( π 2 −t) Fundamental Identities tant=[/itex] sin t cos t sec t= 1 cos t csc t= 1 sin t cot t= 1 tan t = cos t sin t Sum and Difference Identities cos(α+β)=cosαcosβsinαsinβ[/itex] cos(α−β)=cos α cos β+sin α sin β sin(α+β)=sin α cos β+cos α sin β sin(α−β)=sin α cos β−cos α sin β tan(α+β)= tan α+tan β 1−tan α tan β tan(α−β)= tan α−tan β 1+tan α tan β Double-Angle Formulas sin(2θ)=2sinθcosθ[/itex] cos(2θ)= cos 2 θ− sin 2 θ cos(2θ)=1−2  sin 2 θ cos(2θ)=2  cos 2 θ−1 tan(2θ)= 2 tan θ1− tan 2 θ Half-Angle Formulas sin[/itex] α 2 =± 1−cos α 2 cos  α 2 =± 1+cos α 2 tan  α 2 =± 1−cos α 1+cos α tan  α 2 = sin α 1+cos α tan  α 2 =1−cos α sin α Reduction Formulas sin[/itex] 2 θ= 1−cos( 2θ ) 2 cos 2 θ= 1+cos( 2θ ) 2 tan 2 θ= 1−cos( 2θ ) 1+cos( 2θ ) Product-to-Sum Formulas cosαcosβ=[/itex] 1 2 [ cos(α−β)+cos(α+β) ] sin α cos β= 1 2 [ sin(α+β)+sin(α−β) ] sin α sin β= 1 2 [ cos(α−β)−cos(α+β) ] cos α sin β= 1 2 [ sin(α+β)−sin(α−β) ] Sum-to-Product Formulas sinα+sinβ=2sin([/itex] α+β 2 ) cos( α−β 2 ) sin α−sin β=2 sin( α−β 2 ) cos( α+β 2 ) cos α−cos β=−2 sin( α+β 2) sin( α−β 2 ) cos α+cos β=2 cos( α+β 2 ) cos( α−β 2 ) Law of Sines sinα[/itex] a = sin β b = sin γ c a sin α = b sin β = c sin γ Law of Cosines a[/itex] 2 = b 2 + c 2 −2bc cos α b 2 = a 2 + c 2 −2ac cos β c 2 = a 2 + b 2 −2ab cos γ