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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 lemniscate functions sl and cl can be defined as the solution to the initial value problem: [5] = (+ ) , = (+ ) , =, =, or equivalently as the inverses of an elliptic integral, the Schwarz–Christoffel map from the complex unit disk to a square with corners {,,,}: [6]
For example, sulfuric acid (H 2 SO 4) is a diprotic acid. Since only 0.5 mol of H 2 SO 4 are needed to neutralize 1 mol of OH −, the equivalence factor is: f eq (H 2 SO 4) = 0.5. If the concentration of a sulfuric acid solution is c(H 2 SO 4) = 1 mol/L, then its normality is 2 N. It can also be called a "2 normal" solution.
For example, if there are 10 grams of salt (the solute) dissolved in 1 litre of water (the solvent), this solution has a certain salt concentration . If one adds 1 litre of water to this solution, the salt concentration is reduced. The diluted solution still contains 10 grams of salt (0.171 moles of NaCl).
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 ...
BSS (ophthalmic irrigation solution) (produced by Alcon) . Composition per 1 mL: sodium chloride (NaCl) 6.4 mg, potassium chloride (KCl) 0.75 mg, calcium chloride dihydrate (CaCl 2 ·2H 2 O) 0.48 mg, magnesium chloride hexahydrate (MgCl 2 •6H 2 O) 0.3 mg, sodium acetate trihydrate (C 2 H 3 NaO 2 ·3H 2 O) 3.9 mg, sodium citrate dihydrate (C 6 H 5 Na 3 O 7 ·2H 2 O) 1.7 mg, sodium hydroxide ...
A significant improvement is to use the following modification to the above, a trick (due to Singleton [2]) often used to generate trigonometric values for FFT implementations: c 0 = 1 s 0 = 0 c n+1 = c n − (α c n + β s n) s n+1 = s n + (β c n − α s n) where α = 2 sin 2 (π/N) and β = sin(2π/N).
Cobalt(II) thiocyanate is an inorganic compound with the formula Co(SCN) 2. [1] The anhydrous compound is a coordination polymer with a layered structure. The trihydrate, Co(SCN) 2 (H 2 O) 3, is a isothiocyanate complex used in the cobalt thiocyanate test (or Scott test) for detecting cocaine. The test has been responsible for widespread false ...