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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]
The 0 °C isotherm under normal conditions. The freezing level, or 0 °C (zero-degree) isotherm, represents the altitude in which the temperature is at 0 °C (the freezing point of water) in a free atmosphere (i.e. allowing reflection of the sun by snow, icing conditions, etc.).
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.
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.
Madhava's sine table is the table of trigonometric sines constructed by the 14th century Kerala mathematician-astronomer Madhava of Sangamagrama (c. 1340 – c. 1425). The table lists the jya-s or Rsines of the twenty-four angles from 3.75 ° to 90° in steps of 3.75° (1/24 of a right angle , 90°).
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. ...
Quadrant 1 (angles from 0 to 90 degrees, or 0 to π/2 radians): All trigonometric functions are positive in this quadrant. Quadrant 2 (angles from 90 to 180 degrees, or π/2 to π radians): Sine and cosecant functions are positive in this quadrant.