Question for Trig Gurus

[Thomas Klemm]

(08-01-2022 02:36 PM)Albert Chan Wrote:  … because there is no argument reduction.

That's always a good thing.
I just mentioned that here:

(07-31-2022 11:08 AM)Thomas Klemm Wrote:  Of course we could use this formula again and then the above method to calculate \(\tan^{-1}(x)\):

\(
\begin{align}
\tan \frac{x}{2} &= \sqrt{\frac{1-\cos x}{1+\cos x}} \\
\end{align}
\)

We just have to keep in mind that we have to multiply the result by the factor \(2\).

---

We can also have a look at the manual of the Texas Instruments SR-16 where we can find the following formulas:

Inverse Trigonometric Functions

Arc Sine

\(
\begin{matrix}
\arcsin a = \left[ \left( -\frac{9}{20} a^2 + 1 \right)^{-1} \times 10 + 17 \right] \frac{a}{27} & 0 < a < \frac{1}{2}
\end{matrix}
\)

0.00:  0.000000 ( 0.000000)
0.05:  2.865984 ( 2.865984)
0.10:  5.739170 ( 5.739170)
0.15:  8.626925 ( 8.626927)
0.20: 11.536951 (11.536959)
0.25: 14.477471 (14.477512)
0.30: 17.457448 (17.457603)
0.35: 20.486835 (20.487315)
0.40: 23.576884 (23.578178)
0.45: 26.740526 (26.743684)
0.50: 29.992861 (30.000000)

For greater accuracy

\(
\begin{matrix}
\arcsin a = \left\{ \left[ \left(-\frac{25}{42} a^2 + 1 \right)^{-1} \times 189 + 61 \right] \frac{a^2}{1500} + 1 \right \} a
\end{matrix}
\)

0.00:  0.000000 ( 0.000000)
0.05:  2.865984 ( 2.865984)
0.10:  5.739170 ( 5.739170)
0.15:  8.626927 ( 8.626927)
0.20: 11.536959 (11.536959)
0.25: 14.477511 (14.477512)
0.30: 17.457598 (17.457603)
0.35: 20.487294 (20.487315)
0.40: 23.578102 (23.578178)
0.45: 26.743443 (26.743684)
0.50: 29.999316 (30.000000)

Arc Tangent

\(
\begin{matrix}
\arctan a = \left[ \left( \frac{3a^2}{5} + 1 \right)^{-1} \times 5 + 4 \right] \frac{a}{9} & 0 < a < \frac{1}{2}
\end{matrix}
\)

0.00:  0.000000 ( 0.000000)
0.05:  2.862405 ( 2.862405)
0.10:  5.710593 ( 5.710593)
0.15:  8.530768 ( 8.530766)
0.20: 11.309948 (11.309932)
0.25: 14.036315 (14.036243)
0.30: 16.699491 (16.699244)
0.35: 19.290736 (19.290046)
0.40: 21.803065 (21.801409)
0.45: 24.231287 (24.227745)
0.50: 26.571956 (26.565051)

For greater accuracy

\(
\begin{matrix}
\arctan a = \left\{ \left[ \left(\frac{5 a^2}{7} + 1 \right)^{-1} \times 21 + 4 \right] \frac{a^2}{-75} + 1 \right \} a
\end{matrix}
\)

0.00:  0.000000 ( 0.000000)
0.05:  2.862405 ( 2.862405)
0.10:  5.710593 ( 5.710593)
0.15:  8.530766 ( 8.530766)
0.20: 11.309932 (11.309932)
0.25: 14.036242 (14.036243)
0.30: 16.699236 (16.699244)
0.35: 19.290014 (19.290046)
0.40: 21.801309 (21.801409)
0.45: 24.227475 (24.227745)
0.50: 26.564407 (26.565051)