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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. The reciprocal identities arise as ratios of sides in the triangles where this unit line is no longer the hypotenuse.
Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their corresponding side lengths are proportional.. In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) [1] are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
Point P(x,y) on the circle of unit radius at an obtuse angle θ > π/2 Sine function on unit circle (top) and its graph (bottom) The unit circle centered at the origin in the Euclidean plane is defined by the equation: [2] + = Given an angle θ, there is a unique point P on the unit circle at an anticlockwise angle of θ from the x-axis, and ...
A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods of the function. Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry , i.e. a function f is periodic with period P if the graph of f is invariant under translation ...
Scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses). [51] Most allow a choice of angle measurement methods: degrees, radians, and sometimes gradians. Most computer programming languages provide function libraries that include the trigonometric functions. [52]
The six trigonometric functions are defined for every real number, except, for some of them, for angles that differ from 0 by a multiple of the right angle (90°). Referring to the diagram at the right, the six trigonometric functions of θ are, for angles smaller than the right angle:
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The two solutions to this equation are (−1, 0) and (cos φ, sin φ). This allows us to write the latter as rational functions of t (solutions are given below). The parameter t represents the stereographic projection of the point (cos φ, sin φ) onto the y-axis with the center of projection at (−1, 0).