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A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
A tangent line t to a circle C intersects the circle at a single point T. For comparison, secant lines intersect a circle at two points, whereas another line may not intersect a circle at all. This property of tangent lines is preserved under many geometrical transformations, such as scalings, rotation, translations, inversions, and map ...
Sometimes the slopes of the left and right tangent lines are equal, so the tangent lines coincide. This is true, for example, for the curve y = x 2/3, for which both the left and right derivatives at x = 0 are infinite; both the left and right tangent lines have equation x = 0.
If (x, y) are the Cartesian coordinates of a point, then (−x, y) are the coordinates of its reflection across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, (x, −y) are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the ...
Radial lines (those running through the pole) are represented by the equation =, where is the angle of elevation of the line; that is, = , where is the slope of the line in the Cartesian coordinate system. The non-radial line that crosses the radial line = perpendicularly at the point (,) has the equation = ().
The Cartesian (x′,y′) axes are related to the rotated graticule in the same way that the axes (x,y) axes are related to the standard graticule. The tangent transverse Mercator projection defines the coordinates (x′,y′) in terms of −λ′ and φ′ by the transformation formulae of the tangent Normal Mercator projection:
A conformal map acting on a rectangular grid. Note that the orthogonality of the curved grid is retained. While vector operations and physical laws are normally easiest to derive in Cartesian coordinates, non-Cartesian orthogonal coordinates are often used instead for the solution of various problems, especially boundary value problems, such as those arising in field theories of quantum ...
The tangent line to the unit circle at the point A, is perpendicular to , and intersects the y - and x-axes at points = (,) and = (,). The coordinates of these points give the values of all trigonometric functions for any arbitrary real value of θ in the following manner.