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Sexagesimal, also known as base 60, [1] is a numeral system with sixty as its base.It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.
6: 60° The sextant was the unit used by the Babylonians, [26] [27] The degree, minute of arc and second of arc are sexagesimal subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the angle of the equilateral triangle or is 1 / 6 turn. 1 Babylonian unit = 60° = π /3 rad ≈ 1.047197551 ...
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. By symmetry, the bisected side is half of the side of the equilateral triangle, so one concludes sin ( 30 ∘ ) = 1 / 2 {\displaystyle \sin(30^{\circ ...
Set square, shaped as 30° - 60° - 90°° triangle The side lengths of a 30°–60°–90° triangle 30° - 60° - 90° right triangle of hypotenuse length 1. This is a triangle whose three angles are in the ratio 1 : 2 : 3 and respectively measure 30° ( π / 6 ), 60° ( π / 3 ), and 90° ( π / 2 ).
Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit, and further subdivided the latter into 60 parts following their sexagesimal numeric system. [7] [8] The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle ...
A method to solve such problems is to consider the rate of change of the angle in degrees per minute. The hour hand of a normal 12-hour analogue clock turns 360° in 12 hours (720 minutes) or 0.5° per minute. The minute hand rotates through 360° in 60 minutes or 6° per minute. [1]
A disadvantage is that the common angles of 30° and 60° in geometry must be expressed in fractions (as 33 + 1 / 3 ... π / 3 or 𝜏 / 6 rad 60°
Theorem: An angle of measure θ may be trisected if and only if q(t) = 4t 3 − 3t − cos(θ) is reducible over the field extension Q(cos(θ)). The proof is a relatively straightforward generalization of the proof given above that a 60° angle is not trisectible. [6]