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ss_6_5 - Section 6.5 Inverse trigonometric Functions None...

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Section 6.5 Inverse trigonometric Functions None of the six trigonometric functions are one-to-one functions. (The graphs of the trig functions do not satisfy the horizontal line test, for example.) Since the trig functions are not one-to-one functions, they do not have inverse functions. However, we can define inverse functions for trig function on restricted domains, where they are one-to-one functions. One important reason for restricting domains is to a ensure unique solution to equations like sin x = . 5. The graphs of six trigonometric functions are shown below. One can see from the graphs that these functions are not one-to-one. sin x ,cos x ,tan x –1 –0.5 0 0.5 1 2 24681 01 2 t –1 –0.5 0.5 1 2 0 1 2 t –10 –8 –6 –4 –2 0 2 4 6 8 10 y 2 0 1 2 t csc x ,sec x ,cot x –3 –2 –1 0 1 2 3 y 2 2 t –10 –8 –6 –4 –2 0 2 4 6 8 10 y 2 0 1 2 t –3 –2 –1 0 1 2 3 y 2 0 1 2 t We truncate the domains of the trigonometric functions and form new one-to-one (representative) functions. These new functions are called Sin, Cos, Tan, Csc, Sec, Cot. (Note the names of these functions begin with capital letters.) The graphs of the restricted trig functions are shown below. Sin, Cos, Tan –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 y –1.5 –1 –0.5 0.5 1 1.5 t –1 –0.8 –0.6 –0.4 –0.2 0 0.2 0.4 0.6 0.8 1 y 0.5 1 1.5 2 2.5 3 t –3 –2 –1 0 1 2 3 y –1.5 –1 –0.5 0.5 1 1.5 t Csc, Sec, Cot –10 –8 –6 –4 –2 0 2 4 6 8 10 y –1.5 –1 –0.5 0.5 1 1.5 t –10 –8 –6 –4 –2 0 2 4 6 8 10 y 0.5 1 1.5 2 2.5 3 t –3 –2 –1 0 1 2 3 y 0.5 1 1.5 2 2.5 3 t 1
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Since Sin, Cos, Tan, Csc, Sec, Cot are one-to-one functions, they have inverse functions. We denote the inverse functions by Sin - 1 ,Co s - 1 ,T an - 1 ,C sc - 1 ,Sec - 1 t - 1 or by arcsin, arccos, arctan, arccsc, arcsec, arccot.
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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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ss_6_5 - Section 6.5 Inverse trigonometric Functions None...

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