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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.
The other four trigonometric functions (tan, cot, sec, csc) can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator. It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle in radians. [ 5 ]
1.5.3 Tangent and cotangent. 1.6 Double-angle identities. 1.7 Half-angle identities. 1.8 Miscellaneous – the triple tangent identity.
hyperbolic tangent " tanh" (/ ˈ t æ ŋ, ˈ t æ n tʃ, ˈ θ æ n /), [5] hyperbolic cotangent " coth" (/ ˈ k ɒ θ, ˈ k oʊ θ /), [6] [7] hyperbolic secant " sech" (/ ˈ s ɛ tʃ, ˈ ʃ ɛ k /), [8] hyperbolic cosecant " csch" or "cosech" (/ ˈ k oʊ s ɛ tʃ, ˈ k oʊ ʃ ɛ k / [3]) corresponding to the derived trigonometric functions ...
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [ 32 ] With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines . [ 33 ]
All derivatives of circular trigonometric functions can be found from those of sin(x) and cos(x) by means of the quotient rule applied to functions such as tan(x) = sin(x)/cos(x). Knowing these derivatives, the derivatives of the inverse trigonometric functions are found using implicit differentiation.
In this way, this trigonometric identity involving the tangent and the secant follows from the Pythagorean theorem. The angle opposite the leg of length 1 (this angle can be labeled φ = π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity ...
Quadrant 3 (angles from 180 to 270 degrees, or π to 3π/2 radians): Tangent and cotangent functions are positive in this quadrant. Quadrant 4 (angles from 270 to 360 degrees, or 3π/2 to 2π radians): C osine and secant functions are positive in this quadrant.