HP -15C Advanced Functions Handbook page 187

Appendix: Accuracy of Numerical Calculations 185
|TRIG|(:c) = trig(xTr/p)
to within ±0.6 units in its 10th significant digit.
This formula has important practical implications:
• Since rr/p = 1 - 2.0676... X lO'
13
//) = 0.9999999999999342...,
the value produced by [TRIG \(x) differs from trig(x) by no more
than can be attributed to two perturbations: one in the 10th
significant digit of the output trig(o;), and one in the 13th
significant digit of the input x.
If x has been calculated and rounded to 10 significant digits,
the error inherited in its 10th significant digit is probably
orders of magnitude bigger than [TRIG |'s second perturbation
in x's 13th significant digit, so this second perturbation can be
ignored unless x is regarded as known or calculated exactly.
• Every trigonometric identity that does not explicitly involve n
is satisfied to within roundoff in the 10th significant digit of
the calculated values in the identity. For instance,
sin
2
(x) + cos
2
(x) = 1, so (Q3IN](*))
2
+ (fcOSlQc))
2
= 1
sin(;c)/cos(:x;) = tan(x), so| SIN \(x)/\COS\(x) = I TANK*)
with each calculated result correct to nine significant digits
for all x. Note that | C O S \ ( x ) vanishes for no value of x
representable exactly with just 10 significant digits. And if 2x
can be calculated exactly given x,
sin(2x) = 2sin(*)cos(*), so [SJN](2x) = 2 [SIN](x)[cOS](%)
to nine significant digits. Try the last identity for x = 52174
radians on the HP-15C:
[siN](2;c) = -0.00001100815000,
2[SII\n(x)rcOSl(a:) = -0.00001100815000.
Note the close agreement even though for this x, sin(2^;) =
2sin(;t)cos(x) = -0.0000110150176... disagrees with [§JN](2x) in
its fourth significant digit. The same identities are satisfied by
| T R I G | ( x ) values as by trig(jc) values even though [TRIG \(x) and
trig(x) may disagree.
• Despite the two kinds of errors in [TRIG |, its computed values
preserve familiar relationships wherever possible:
• Sign symmetry: |COS|(-*) = |COS|(x)