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The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained.
Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
The sine, cosine, and tangent ... 30°, 45°, 60° and 90° follow the pattern with n = 0, 1 ... the specific values for each identity are summarized in this table:
The following table shows the special value of each input for both sine and cosine with the domain between < ... Exact Decimal Exact Decimal 0° 0 ... 60° 1 / 3 ...
Abu al-Wafa had sine tables in 0.25° increments, to 8 decimal places of accuracy, and accurate tables of tangent values. [16] He also made important innovations in spherical trigonometry [17] [18] [19] The Persian polymath Nasir al-Din al-Tusi has been described as the creator of trigonometry as a mathematical discipline in its own right.
The exact formula is ... The 1 in 60 rule used in air ... where the values for sin(0.75) and cos(0.75) are obtained from trigonometric table. The result is accurate ...
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These two starting trigonometric values are usually computed using existing library functions (but could also be found e.g. by employing Newton's method in the complex plane to solve for the primitive root of z N − 1). This method would produce an exact table in exact arithmetic, but has errors in finite-precision floating-point arithmetic