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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
A dark star, therefore, has a rarefied atmosphere of "visiting particles", and this ghostly halo of matter and light can radiate, albeit weakly. Also as faster-than-light speeds are possible in Newtonian mechanics, it is possible for particles to escape. Radiation effects A dark star may emit indirect radiation as described above.
In April 2023, a study investigated four extremely redshifted objects discovered by the James Webb Space Telescope. [5] Their study suggested that three of these four, namely JADES-GS-z13-0, JADES-GS-z12-0, and JADES-GS-z11-0, are consistent with being point sources, and further suggested that the only point sources which could exist in this time and be bright enough to be observed at these ...
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.
Then multiplying the numerator and denominator inside the square root by (1 + cos θ) and using Pythagorean identities leads to: = + . Also, if the numerator and denominator are both multiplied by (1 - cos θ), the result is:
The mathematics of trigonometry and exponentials are related but not exactly the same; exponential notation emphasizes the whole, whereas cis x and cos x + i sin x notations emphasize the parts. This can be rhetorically useful to mathematicians and engineers when discussing this function, and further serve as a mnemonic (for cos + i sin ).
If vectors u and v have direction cosines (α u, β u, γ u) and (α v, β v, γ v) respectively, with an angle θ between them, their units vectors are ^ = + + (+ +) = + + ^ = + + (+ +) = + +. Taking the dot product of these two unit vectors yield, ^ ^ = + + = , where θ is the angle between the two unit vectors, and is also the angle between u and v.
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.