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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.
Dark stars and black holes both have a surface escape velocity equal or greater than lightspeed, and a critical radius of r ≤ 2M. However, the dark star is capable of emitting indirect radiation – outward-aimed light and matter can leave the r = 2M surface briefly before being recaptured, and while outside the critical surface, can interact ...
Dark star (Newtonian mechanics), a star that has a gravitational pull strong enough to trap light under Newtonian gravity Dark star (dark matter), a star heated by annihilation of dark matter particles within it
[1] [2] [3] In the unlikely event that dark stars have endured to the modern era, they could be detectable by their emissions of gamma rays, neutrinos, and antimatter and would be associated with clouds of cold molecular hydrogen gas that normally would not harbor such energetic, extreme, and rare particles. [4] [2]
[1] [2] One reason for this is that they can greatly simplify differential equations that do not need to be answered with absolute precision. There are a number of ways to demonstrate the validity of the small-angle approximations. The most direct method is to truncate the Maclaurin series for each of the trigonometric functions.
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
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 ...
which by the Pythagorean theorem is equal to 1. This definition is valid for all angles, due to the definition of defining x = cos θ and y sin θ for the unit circle and thus x = c cos θ and y = c sin θ for a circle of radius c and reflecting our triangle in the y-axis and setting a = x and b = y.