Use the Trig Powers field of the Options > Mode Settings > Simplification command19_L5FP to control the transformation of powers of sines into cosines and vice versa.  The Trig Powers field provides the following options:

•        Select the Sines option to transform powers of cosines into sines when simplifying expressions.

•        Select the Cosines option to transform powers of sines into cosines 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 powers setting.


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

Trigpower := direction

where direction is Auto, Sines, or Cosines.  The trig power 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 Cosines direction is selected, the following transformations are applied:

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

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

if m is an integer.  For example, if the Cosines direction is selected,

SIN(x)^4

simplifies to

      4           2    
COS(x)  - 2·COS(x)  + 1


When the Sines direction is selected, the following transformations are applied:

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

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

if m is an integer.  For example, if the Sines direction is selected,

COS(x)^4 - 2·COS(x)^2 + 1

simplifies to

      4
SIN(x) 


When the Auto direction is selected, only the above 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 Sines or Cosines trig power direction enables a simplification that does not occur in the Auto direction.  On the other hand, selecting the Sines or Cosines direction may produce huge results and take a long time to simplify.


Other Options > Mode Settings > Simplification command19_L5FP fields

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