Use the Trigonometry field of the Options > Mode Settings > Simplification command19_L5FP to control the application of the angle-sum and multiple-angle trigonometric transformations.  The Trigonometry field provides the following options:

•        Select the Collect option to collect products and powers of trig functions into trig functions of angle sums and multiple angles when simplifying expressions.

•        Select the Expand option to expand trig functions of angle sums and multiple angles into products and powers of trig functions when simplifying expressions.

•        Select the Auto option to apply the transformations in such a way as to produce the most compact and attractive form when simplifying expressions.  Auto is the factory default trig mode setting.


When a new trigonometry transform direction is selected, this command generates an expression of the form

Trigonometry := direction

where direction is Auto, Collect, or Expand.  The trigonometry transform direction can also be changed by entering on the expression entry line an expression of the above form using upper and lower case exactly as shown.


When the Expand trigonometry direction is selected, the following transformations are applied:

SIN(z+w)  ⇒  SIN(z)·COS(w) + COS(z)·SIN(w)

COS(z+w)  ⇒  COS(z)·COS(w) - SIN(z)·SIN(w)

SIN(m·z)  ⇒  2·COS(z)·SIN((m-1)·z) + SIN((m-2)·z)

COS(m·z)  ⇒  -2·SIN(z)·SIN((m-1)·z) + COS((m-2)·z)

if m is a positive integer greater than 1.  Also, trig functions with arguments of the form π/n, where n is a power of 2 times 1, 3, 5, or 15, are expanded into algebraic numbers using radicals.  For example, if the Expand direction is selected,

SIN(2·x + y)

simplifies to

        2                                          
2·COS(x) ·SIN(y) + 2·SIN(x)·COS(x)·COS(y) - SIN(y) 

and

COS(2·pi/15)

simplifies to

 ⎛ 15     3·√5 ⎞    √5     1 
√⎜———— - ——————⎟ + ———— + ———
 ⎝ 32      32  ⎠     8     8 


When the Collect trigonometry direction is selected, the following transformations are applied:

                    COS(z-w) - COS(z+w) 
SIN(z)·SIN(w)  ⇒  —————————————————————
                             2          

                    COS(z-w) + COS(z+w) 
COS(z)·COS(w)  ⇒  —————————————————————
                             2          

                    SIN(z-w) + SIN(z+w) 
SIN(z)·COS(w)  ⇒  —————————————————————
                             2          

      m       1 - COS(2·z)        m-2
SIN(z)   ⇒  ——————————————·SIN(z)   
                    2                

      m       1 + COS(2·z)        m-2
COS(z)   ⇒  ——————————————·COS(z)   
                    2                

a·SIN(z) + b·COS(z)  ⇒                      

           2    2      π·SIGN(b)         a       
        √(a  + b )·SIN(————————— - ATAN(———) + z)
                           2             b       


if m is a positive integer greater than 1.  For example, if the Collect direction is selected,

2·COS(x)^2·SIN(y) + 2·SIN(x)·COS(x)·COS(y) - SIN(y)

simplifies to

SIN(2·x + y)


When the Auto trigonometry direction is selected, only those trigonometric transformations that clearly yield a simplification are applied.  Usually this results in the most compact and attractive form of trigonometric expressions.  However, you may find examples where selecting the Collect or Expand trigonometry direction enables a simplification that does not occur in the Auto direction.  On the other hand, selecting the Collect or Expand direction may produce huge results and take a long time to simplify.


Other Options > Mode Settings > Simplification command19_L5FP fields

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