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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.
The yields in the table sum to only 45.5522%, including 34.8401% which have half-lives greater than one year: ... 28.9 y: Source of much of the decay heat together ...
To find P d (x), have a total of x dots composing the target shape. P d (x) then equals the total number of dots in the shape. A 0-dimensional triangle is a point and a 1-dimensional triangle is simply a line, and therefore P 0 (x) = 1 and P 1 (x) = x, which is the sequence of natural numbers. The number of dots in each layer corresponds to P d ...
1 Bq = 1 s −1. A special name was introduced for the reciprocal second (s −1) to represent radioactivity to avoid potentially dangerous mistakes with prefixes.For example, 1 μs −1 would mean 10 6 disintegrations per second: (10 −6 s) −1 = 10 6 s −1, [4] whereas 1 μBq would mean 1 disintegration per 1 million seconds.
Just as for the trigonometric functions, values of the lemniscate functions can be computed for divisions of the lemniscate into n parts of equal length, using only basic arithmetic and square roots, if and only if n is of the form = where k is a non-negative integer and each p i (if any) is a distinct Fermat prime.
Moreover, as the derivative of f(x) evaluates to ln(b) b x by the properties of the exponential function, the chain rule implies that the derivative of log b x is given by [35] [37] = . That is, the slope of the tangent touching the graph of the base- b logarithm at the point ( x , log b ( x )) equals 1/( x ln( b )) .
In a similar manner it may be shown that the sum of the first Fibonacci numbers up to the n-th is equal to the (n + 2)-th Fibonacci number minus 1. [34] In symbols: = = + This may be seen by dividing all sequences summing to + based on the location of the first 2.
Euclid's formula for Pythagorean triples and the inverse relationship t = y / (x + 1) mean that, except for (−1, 0), a point (x, y) on the circle is rational if and only if the corresponding value of t is a rational number. Note that t = y / (x + 1) = b / (a + c) = n / m is also the tangent of half of the angle that is opposite the triangle ...