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Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in differential geometry and has been extensively generalized; see Tangent space. The word "tangent" comes from the Latin tangere, "to touch".
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent functions. Their reciprocals are respectively the cosecant , the secant , and the cotangent functions, which are less used.
Trigonometric functions were among the earliest uses for mathematical tables. [48] Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. [49] Slide rules had special scales for trigonometric functions. [50]
These identities are useful whenever expressions involving trigonometric functions need to be simplified. An important application is the integration of non-trigonometric functions: a common technique involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
The most direct method is to truncate the Maclaurin series for each of the trigonometric functions. Depending on the order of the approximation , cos θ {\displaystyle \textstyle \cos \theta } is approximated as either 1 {\displaystyle 1} or as 1 − 1 2 θ 2 {\textstyle 1-{\frac {1}{2}}\theta ^{2}} .
Signs of trigonometric functions in each quadrant. All Students Take Calculus is a mnemonic for the sign of each trigonometric functions in each quadrant of the plane. The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving counterclockwise through quadrants 2 to 4.
In trigonometry, the law of tangents or tangent rule [1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a , b , and c are the lengths of the three sides of the triangle, and α , β , and γ are the angles opposite those three respective sides.
Geometry is used extensively in trigonometry. Angle – the angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane.