SIMILAR RIGHT TRIANGLE  (SRT) TABLE OF MULTIPLICATIONS

Essentially number-line multiplication and division accomplished

by construction with only the compass and straightedge

 

sin = A/H = 1/csc        csc=H/A = 1/sin          sin G = cos OPP     csc G = sec OPP

cos= B/H = 1/sec         sec=H/B = 1/cos        cos G = sin OPP     sec G = csc OPP

tan= A/B = 1/cot         cot=B/A = 1/tan          tan G = cot OPP     cot G = tan OPP

 

S=sine CS=cosecant C=cosine SC=secant T=tangent CT=cotangent

 

If A=X            H=X * CS     H= X/S       H=B * SC      H=B/C    

                        B=X * CT     B=X/T        B= H * C       B=H/SC

If B=X,            H=X * SC     H=X/C        H= B * SC     H= B/C       

                        A=X * T       A=X/CT      A=H * S         A=H/CS  

If H=X,           A=X * S       A=X/CS      A=H * S          A=H/CS  

                        B=X * C       B=X/SC      B=H * C          B=H/SC

 

and      X * sin = X/csc           X * csc = X/sin

            X * cos = X/sec          X * sec = X/cos

            X * tan = X/cot           X * cot = X/tan

Conversion to functions of sides: (S1=side 1)  (this section optional)

S1=A1             S2=B1             S3=H1

S4=A2             S5=B2             S6=H2

 

A2...                B2=(A2*B1)/A1         H2=(A2*H1)/A1

B2...                A2=(B2*A1)/B1         H2=(B2*H1)/B1

H2...                A2=(H2*A1)/H1         B2=(H2*B1)/H1

S4=A2...          B2=(A2*B1)/A1         H2=(A2*H1)/A1

S5=B2...          A2=(B2*A1)/B1         H2=(B2*H1)/B1

S6=H2...          A2=(H2*A1)/H1         B2=(H2*B1)/H1

 

S4=A2...          S5=(S4*S2)/S1           S6=(S4*S3)/S1

S5=B2...          S4=(S5*S1)/S2           S6=(S5*S3)/S2

S6=H2...          S4=(S6*S1)/S3           S5=(S6*S2)/S3

 

 

if S4=A2...      S[x+1]=(S[x]*S[x-2])/S[x-3]     S[x+2]=(S[x]*S[x-1])/S[x-3]

if S5=B2...      S[x]=(S[x+1]*S[x-3])/S[x-2]    S[x+2]=(S[x+1]*S[x-1])/S[x-2]

if S6=Hy2...    S[x]=(S[x+2]*S[x-3])/S[x-1]    S[x+1]=(S[x+2]*S[x-2])/S[x-1]

 

(this section clarifies)

To restate briefly the reality of re-constructing right triangle angles and sides:

Construct A, and:  A=A                      B=A*cotangent                       H=A*cosecant

Construct B, and:  A=B*tangent        B=B                                         H=B*secant

Construct H, and:  A=H*sine             B=H*cosine                            H=H

 

Divide a line by a ratio (or a line value) -- similar right triangle [table]:

This construction procedure is similar to the multiplication construction except for the simple difference that the ratio is the inverse of the multiplier, which results, obviously, in multiplication by the inverse ratio or fraction.  Invert the ratio 3 to 1 to get the ratio 1 to 3 or one third, and so forth.  In this example, X * (1/3) is of course X divided by 3.