Harold S Trig Proofs

Harold’s Trig Proofs

Cheat Sheet

26 April 2016

Proof of Pythagorean Identities
Proof

Given / x2+y2=r2
r=1
sinθ=oppositehypotenuse=yr=y1=y
cosθ=adjacenthypotenuse=xr=x1=x
Substitute and Simplify / sin2θ+cos2θ=12
Formula / sin2θ+cos2θ=1 [1]
Proof
Given / sin2θ+cos2θ=1 [1]
Divide by cos2θ, then Simplify / sin2θcos2θ+cos2θcos2θ=1cos2θ
Formula / tan2θ+1=sec2θ [2]
Proof
Given / sin2θ+cos2θ=1 [1]
Divide by sin2θ, then Simplify / sin2θsin2θ+cos2θsin2θ=1sin2θ
Formula / 1+cot2θ=csc2θ [3]
Proof of Sum and Difference Formulas
Trig Sum and Difference Formulas / sinα±β=sinαcosβ±cosαsinβ
cosα±β=cosαcosβ∓sinαsinβ
Proof Diagram /
Proof of sinα±β
Prove Sum
Given / sinα+β=EDDA=oppositehypotenuse
Alternate interior angles are congruent / α=∠CAB=∠HFA=∠HDF
Tallest vertical line / ED=GF+HD
Substitute, then divide and multiply
by AF & FD / sinα+β=EDAD=GFAD+HDAD=GFAF AFAD+HDFD FDAD
Convert back to trig formulas / sinα+β=sinαcosβ+cosαsinβ [4]
Prove Difference
Replace +β with -β / cos-β=cosβ
sin-β=-sinβ
Simplify / sinα-β=sinαcosβ-cosαsinβ [5]
General Formula [4+5] / sinα±β=sinαcosβ±cosαsinβ [6]
Proof of cosα±β
Prove Sum
Given / cosα+β=AEAD=adjacenthypotenuse
Longest horizontal line / EA=GA-FH
Substitute, then divide and multiply
by AF & DF / cosα+β=EAAD=GAAD-FHAD=GAAF AFAD+FHDF DFAD
Convert back to trig formulas / cosα+β=cosαcosβ-sinαsinβ [7]
Prove Difference
Replace +β with -β / cos-β=cosβ
sin-β=-sinβ
Simplify / cosα-β=cosαcosβ+sinαsinβ [8]
General Formula [7+8] / cosα±β=cosαcosβ∓sinαsinβ [9]
Proof of tanα±β
Prove Sum and Difference
Given / tanα±β=sinα±βcosα±β
Substitute / sinα±β=sinαcosβ±cosαsinβ [6]
cosα±β=cosαcosβ∓sinαsinβ [9]
Divide by (cosαcosβ), then Simplify / tanα±β=sinαcosβ±cosαsinβcosαcosβ∓sinαsinβ
General Formula / tanα±β=tanα± tanβ1∓ tanα tanβ [10]
Proof of Double Angle Formulas (2θ)
Proof
Given / sinα+β=sinαcosβ+cosαsinβ [4]
Substitute / θ=α=β
Simplify / sinθ+θ=sinθcosθ+cosθsinθ
Formula / sin2θ=2sinθcosθ [14]
Proof
Given / cosα+β=cosαcosβ-sinαsinβ [7]
Substitute / θ=α=β
Simplify / cosθ+θ=cosθcosθ-sinθsinθ
Formula / cos2θ=cos2θ-sin2θ [15]
Proof
Given / cos2θ=cos2θ-sin2θ [15]
sin2θ+cos2θ=1 [1]
Substitute / sin2θ=1-cos2θ
Simplify / cos2θ=cos2θ-1-cos2θ
Formula / cos2θ=2 cos2θ-1 [16]
Proof
Given / cos2θ=cos2θ-sin2θ [15]
sin2θ+cos2θ=1 [1]
Substitute / cos2θ=1-sin2θ
Simplify / cos2θ=1-sin2θ-sin2θ
Formula / cos2θ=1-2sin2θ [17]
Proof
Given / tan2θ=sin2θcos2θ
Substitute / sin2θ=2sinθcosθ [14]
cos2θ=cos2θ-sin2θ [15]
Divide by cos2θ / tan2θ= 2sinθcosθcos2θ-sin2θ
Simplify / tan2θ= 2sinθcosθcos2θcos2θ-sin2θcos2θ
Formula / tan2θ=2 tanθ1-tan2θ [18]
Proof of Half Angle Formulas (θ/2)
Proof
Given / cos2θ=1-2sin2θ [17]
Solve for sin2θ / sin2θ=1-cos2θ2 [19a]
Substitute / θ=θ2
Solve / sin2θ2=1-cosθ2
Formula / sinθ2=±1-cosθ2 [19b]
Proof
Given / cos2θ=2 cos2θ-1 [16]
Solve for cos2θ / cos2θ=1+cos2θ2 [20a]
Substitute / θ=θ2
Solve / cos2θ2=1+cosθ2
Formula / cosθ2=±1+cosθ2 [20b]
Proof
Given / tan2θ=sin2θcos2θ
Substitute / sin2θ=1-cos2θ2 [19a]
cos2θ=1+cos2θ2 [20a]
Simplify / tan2θ=1-cos2θ21+cos2θ2=1-cos2θ1+cos2θ
Substitute / θ=θ2
Solve / tan2θ2=1-cosθ1+cosθ [21a]
Formula / tanθ2=±1-cosθ1+cosθ [21b]
Proof of Cofunction Formulas
Proof
Given / sinα-β=sinαcosβ-cosαsinβ [5]
Substitute / α=π2, β=θ
Simplify / sinπ2-θ=sinπ2cosθ+cosπ2sinθ
Formula / sinπ2-θ=cosθ [22]
Proof
Given / cosα-β=cosαcosβ+sinαsinβ [8]
Substitute / α=π2, β=θ
Simplify / cosπ2-θ=cosπ2cosθ-sinπ2sinθ
Formula / cosπ2-θ=sinθ [23]
Proof
Given / tanθ=sinθcosθ
Substitute / sinπ2-θ=cosθ [22]
cosπ2-θ=sinθ [23]
Simplify / tanπ2-θ=sinπ2-θcosπ2-θ=cosθsinθ=cotθ
Formula / tanπ2-θ=cotθ [24]
Proof
Given / secθ=1cosθ
Substitute / cosπ2-θ=sinθ [23]
Simplify / secπ2-θ=1cosπ2-θ=1sinθ=cscθ
Formula / secπ2-θ=cscθ [25]
Proof
Given / cscθ=1sinθ
Substitute / sinπ2-θ=cosθ [22]
Simplify / cscπ2-θ=1sinπ2-θ=1cosθ=secθ
Formula / cscπ2-θ=secθ [26]
Proof
Given / cotθ=1tanθ
Substitute / tanπ2-θ=cotθ [24]
Simplify / cotπ2-θ=1tanπ2-θ=1cotθ=tanθ
Formula / cotπ2-θ=tanθ [27]