PreCalculus Course Outcome 4 - Inverse Trigonometric Functions PDF

MATH04 PreCalculus Course Outcome 4 Lesson 1: Inverse Trigonometric Functions INVERSE TRIGONOMETRIC FUNCTIONS If 𝑓 is a 𝑜𝑛𝑒 − 𝑡𝑜 − 𝑜𝑛𝑒 function with domain 𝑨 and range 𝑩, then its inverse 𝑓 −1 is the function with domain 𝑩 and range 𝑨 defined by...

MATH04 PreCalculus Course Outcome 4 Lesson 1: Inverse Trigonometric Functions INVERSE TRIGONOMETRIC FUNCTIONS If 𝑓 is a 𝑜𝑛𝑒 − 𝑡𝑜 − 𝑜𝑛𝑒 function with domain 𝑨 and range 𝑩, then its inverse 𝑓 −1 is the function with domain 𝑩 and range 𝑨 defined by 𝑓 −1 𝑥 𝑓 𝑦 =𝑥 INVERSE TRIGONOMETRIC FUNCTIONS For a function to have an inverse, it must be one-to-one. Since the trigonometric functions are not one-to-one, they do not have inverses. It is possible, however, to restrict the domains of the trigonometric functions in such a way that the resulting functions are one-to-one. INVERSE SINE FUNCTION and ITS GRAPH 𝜋 𝜋 𝑦 = sin−1 x 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 x = sin y, 𝑤ℎ𝑒𝑟𝑒 − 1 ≤ 𝑥 ≤ 1 𝑎𝑛𝑑 − ≤𝑦≤. 2 2 y y  1 2 o x o x −    −1 1 − 2 2  𝒚 = sin−𝟏 𝒙 −1 𝒚 = sin 𝒙 − 2    D : − 1 , 1 D : − ,   2 2    R : − ,  R : − 1 , 1  2 2  INVERSE COSINE FUNCTION and ITS GRAPH 𝑦 = cos −1 x 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥 = cos y, 𝑤ℎ𝑒𝑟𝑒 − 1 ≤ 𝑥 ≤ 1 𝑎𝑛𝑑 0 ≤ 𝑦 ≤ 𝜋. y y 1   o x 2 −    2 − 2 2 −1 𝒚 = cos 𝒙 o x −1 1 −𝟏 𝒚 = cos 𝒙 D : 0 ,   D : − 1 , 1 R : - 1, 1  R :  0,   INVERSE TANGENT FUNCTION and ITS GRAPH 𝜋 𝜋 𝑦 = tan−1 x 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑥 = tan y, where − ∞ < 𝑥 < ∞ 𝑎𝑛𝑑 −