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The lemniscate is symmetric to the line connecting its foci F 1 and F 2 and as well to the perpendicular bisector of the line segment F 1 F 2. The lemniscate is symmetric to the midpoint of the line segment F 1 F 2. The area enclosed by the lemniscate is a 2 = 2c 2. The lemniscate is the circle inversion of a hyperbola and vice versa.
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
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.
Trigonometric functions and their reciprocals on the unit circle. All of the right-angled triangles are similar, i.e. the ratios between their corresponding sides are the same. For sin, cos and tan the unit-length radius forms the hypotenuse of the triangle that defines them.
Closed graph theorem [5] — If : is a map from a topological space into a Hausdorff space, then the graph of is closed if : is continuous. The converse is true when Y {\displaystyle Y} is compact .
The y-axis ordinates of A, B and D are sin θ, tan θ and csc θ, respectively, while the x-axis abscissas of A, C and E are cos θ, cot θ and sec θ, respectively. Signs of trigonometric functions in each quadrant. Mnemonics like "all students take calculus" indicates when sine, cosine, and tangent are positive from quadrants I to IV. [8]
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 ...
Another example of a curve with infinite length is the graph of the function defined by f(x) = x sin(1/x) for any open set with 0 as one of its delimiters and f(0) = 0. Sometimes the Hausdorff dimension and Hausdorff measure are used to quantify the size of such curves.