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In contrast, by the Lindemann–Weierstrass theorem, the sine or cosine of any non-zero algebraic number is always transcendental. [4] The real part of any root of unity is a trigonometric number. By Niven's theorem, the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2. [5]
1.1 to 1 ratio of salt to ice. Ice: Calcium chloride hexahydrate-10 1 to 2.5 ratio of salt to ice. Liquid N 2: Ethylene glycol-10 Ice: Acetone-10 1 to 1 ratio of acetone to ice. Liquid N 2: Cycloheptane-12 Dry ice: Benzyl alcohol-15 Dry ice: Ethylene glycol-15 Ice: Sodium chloride-20 1 to 3 ratio of salt to ice. Dry ice: Tetrachloroethylene-22 ...
The quantity 206 265 ″ is approximately equal to the number of arcseconds in a circle (1 296 000 ″), divided by 2π, or, the number of arcseconds in 1 radian. The exact formula is = (″) and the above approximation follows when tan X is replaced by X.
In radians, one would require that 0° ≤ x ≤ π/2, that x/π be rational, and that sin(x) be rational. The conclusion is then that the only such values are sin(0) = 0, sin(π/6) = 1/2, and sin(π/2) = 1. The theorem appears as Corollary 3.12 in Niven's book on irrational numbers. [2] The theorem extends to the other trigonometric functions ...
Boiling point (°C) K b (°C⋅kg/mol) Freezing point (°C) K f (°C⋅kg/mol) Data source; Aniline: 184.3 3.69 –5.96 –5.87 K b & K f [1] Lauric acid: 298.9 44 –3.9 Acetic acid: 1.04 117.9 3.14 16.6 –3.90 K b [1] K f [2] Acetone: 0.78 56.2 1.67 –94.8 K b [3] Benzene: 0.87 80.1 2.65 5.5 –5.12 K b & K f [2] Bromobenzene: 1.49 156.0 6. ...
Comparison of graphs of the parabolas x(180 − x)/8100 and x(180 − x)/9000 with the graph of sin x (with x in degrees) The part of the graph of sin x in the range from 0° to 180° "looks like" part of a parabola through the points (0, 0) and (180, 0). The general form of such a parabola is (). The parabola that also passes through (90, 1 ...
sin(x) cos(x) Degrees Radians Gradians Turns Exact Decimal Exact Decimal 0° 0 0 g: 0 0 0 1 1 30° 1 / 6 π 33 + 1 / 3 g 1 / 12 1 / 2 0.5 0.8660 45° 1 / 4 π: 50 g 1 / 8 0.7071 0.7071 60° 1 / 3 π 66 + 2 / 3 g 1 / 6
Similar to the Kelvin scale, which was first proposed in 1848, [1] zero on the Rankine scale is absolute zero, but a temperature difference of one Rankine degree (°R or °Ra) is defined as equal to one Fahrenheit degree, rather than the Celsius degree used on the Kelvin scale.