Question for Trig Gurus

[Thomas Klemm]

Bhaskara's Sine and Cosine Approximations gives a simple formula to approximate \(\cos^{-1}(x)\) in degrees:

\(\cos^{-1}(x) \approx 180 \sqrt{\frac{1 - x}{4 + x}}\)

Here is a table to compare the results with the expected values in parentheses:

0.00 90.00 (90.00)
0.05 87.18 (87.13)
0.10 84.33 (84.26)
0.15 81.46 (81.37)
0.20 78.56 (78.46)
0.25 75.62 (75.52)
0.30 72.63 (72.54)
0.35 69.58 (69.51)
0.40 66.47 (66.42)
0.45 63.28 (63.26)
0.50 60.00 (60.00)
0.55 56.61 (56.63)
0.60 53.08 (53.13)
0.65 49.38 (49.46)
0.70 45.48 (45.57)
0.75 41.29 (41.41)
0.80 36.74 (36.87)
0.85 31.66 (31.79)
0.90 25.71 (25.84)
0.95 18.09 (18.19)


For improved accuracy we can add a single Newton iteration:

\(\alpha{'} = \alpha + \frac{180}{\pi} \frac{\cos(\alpha) - x}{\sin(\alpha)}\)

This leads to this table with improved accuracy:

0.00 90.0000 (90.0000)
0.05 87.1340 (87.1340)
0.10 84.2608 (84.2608)
0.15 81.3731 (81.3731)
0.20 78.4631 (78.4630)
0.25 75.5225 (75.5225)
0.30 72.5424 (72.5424)
0.35 69.5127 (69.5127)
0.40 66.4218 (66.4218)
0.45 63.2563 (63.2563)
0.50 60.0000 (60.0000)
0.55 56.6330 (56.6330)
0.60 53.1301 (53.1301)
0.65 49.4584 (49.4584)
0.70 45.5731 (45.5730)
0.75 41.4098 (41.4096)
0.80 36.8701 (36.8699)
0.85 31.7886 (31.7883)
0.90 25.8422 (25.8419)
0.95 18.1952 (18.1949)


Example

x = 0.6

0.4
÷
4.6
=
√
×
180
=

53.079104

And then from this to get a better approximation:

cos
-
0.6
÷
53.079104
sin
×
180
÷
\(\pi\)
+
53.079104
=

53.130117 (53.130102)


However it's a bit tedious to reenter the intermediate result twice due to the lack of a storage register.