(12C) Bhaskara's Sine and Cosine Approximations

[Thomas Klemm]

Inverse Sine and Cosine Approximations

Inverse Cosine Function

The approximation for \(\cos(x)\) allows to find an approximation for \(\cos^{-1}(x)\) as well:

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

Program

Code:

01-     01         1
02-     34         x≷y
03-     30         −
04-     43 36    g LSTx
05-     04         4
06-     40         +
07-     10         ÷
08-     43 21    g √x
09-     01         1
10-     08         8
11-     00         0
12-     20         ×
13-     43 33 00 g GTO 00

Examples

In parentheses you can find the correct value.

0
R/S
90.000

0.5
R/S
60.000

1
R/S
0.000

0.5 √x
R/S
44.900 (45)

0.75 √x
R/S
29.867 (30)

5 √x 1 + 4 ÷
R/S
35.871 (36)

Accuracy

For special values (e.g. 0, 0.5, 1) the approximation is exact.
But in general the relative error is within a few ‰.

Inverse Sine Function

To calculate \(\sin^{-1}(x)\) we can simply use:

\(\sin^{-1}(x)=90-\cos^{-1}(x)\)