HP
Prime: Trigonometric CAS Rewrite
Commands
All of the
following examples have CAS Simplification set to Minimum.
Rewrites of
Arcsine and Sine
asin2acos

Toolbox →
4. Rewrite → 5. Sine → 1. asin x → acos x

asin2acos(2*asin(x))
returns
2*(acos(x) + π/2)

asin2atan

Toolbox →
4. Rewrite → 5. Sine → 2. asin x → atan x

asin2atan(2*asin(x))
returns
2*(atan(x/√(x^2+1))

sin2costan

Toolbox →
4. Rewrite → 5. Sine → 3. sin x → cos x tan x

sin2costan(2*sin(x)^2)
returns
2*(cos(x)*tan(x))^2

Rewrites of
Arcosine and Cosine
acos2asin

Toolbox →
4. Rewrite → 6. Cosine → 1. acos x → asin x

acos2asin(acos(x*π/2))
returns
asin(1/2*π*x)+π/2

acos2atan

Toolbox →
4. Rewrite → 6. Cosine → 2. acos x → atan x

acos2atan(acos(x*π/2))
returns
atan( (1/2*π*x)/
(√(1/4) –
√((π*x)^2+4)
)+π/2

cos2sintan

Toolbox →
4. Rewrite → 6. Cosine → 3. cos x → sin x/tan x

cos2sintan(2*cos(x)^2)
Returns
2*(sin(x)/tan(x))^2

Rewrites of
Arctangent and Tangent
atan2asin

Toolbox →
4. Rewrite → 7. Tangent → 1. atan x → asin x

atan2asin(atan(x/3))
returns
asin( (x/3) /
(√(1/9)*√(x^2+9))
simplify gets
asin( (x*√(x^2+9))
/
(x^2+9) )

atan2acos

Toolbox →
4. Rewrite → 7. Tangent → 2. atan x → acos x

atan2acos(atan(x/3))
returns
acos( (x/3) /
(√(1/9)*√(x^2+9))+π/2
simplify gets
( (π2*acos(
(x*√(x^2+9))/(x^2+9)
)/2

tan2sincos

Toolbox →
4. Rewrite → 7. Tangent → 3. tan x → sin x/cos x

tan2sincos(tan(2*x))
returns
sin(2*x)/cos(2*x)

halftan
(sin x,
cos x, tan x) → tan(x/2)

Toolbox →
4. Rewrite → 7. Tangent → 4. halftan

halftan(cos(2*x))
returns
(tan(x)^2+1)/
(tan(x)^2+1)

Other
Commands
tlin
(trigonometric
form of powers) → (trigonometric form of linearized terms)

Toolbox →
4. Rewrite → 8. Trig → 5. tlin

tlin(cos(2*x)^2)
returns
1/2*cos(4*x)+1/2

trig2exp
(sin x,
cos x, tan x) → (exponential form with complex exponentials)

Toolbox →
4. Rewrite → 8. Trig → 8. trig2exp

trig2exp(sin(x/3))
returns
1/(2*i) *
(e^(i*x/3) – 1/e^(i*x/3))

atrig2ln
(asin x,
acos x, atan x) → (logarithm form with complex exponentials)

Toolbox →
4. Rewrite → 8. Trig → 4. atrig2ln

atrig2ln(asin(x/3))
returns
i*ln(√(1/9)*√(x^29)+x/3)+π/2

That is a
list of basic CAS commands for trigonometric rewrites.
Eddie
This blog is
property of Edward Shore, 2016.
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