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inverse_trig_functions

inverse_trig_functions - − √ 3 let y = arctan − √ 3...

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Math 111, Inverse Trigonometric Functions The graph of y = sin x over its domain ( −∞ , ) clearly demonstrates the function is not one-to-one (horizontal line test). However, by restricting the domain of y = sin x to the interval [ π 2 , π 2 ], we have sin x is one-to-one and has an inverse, denoted by arcsin x . Note that the graph of y = arcsin x is the result of reFecting y = sin x for x [ π 2 , π 2 ] about the line y = x . In like manner, the remaining ±ve trigonometric functions have inverses, each with a re- stricted domain (KNOW*). 1. y = arcsin x sin y = x 1 x 1 π 2 y π 2 * 2. y =arccos x cos y = x 1 x 10 y π * 3. y =arctan x tan y = x −∞ ≤ x ≤∞ − π 2 <y< π 2 * 4. y =arccot x cot y = x −∞ ≤ x ≤∞ 0 <y<π 5. y =arcsec x sec y = x | x |≥ 10 y π, y ° = π 2 6. y =arccsc x csc y = x | x |≥ 1 π 2 y π 2 ,y ° =0 Values for inverse trigonometric functions: 1. To ±nd arcsin(1), let y = arcsin(1), then sin y =1for π 2 y π 2 . Therefore y must be the value π 2 so that arcsin(1) = π 2 . 2. To ±nd arctan(
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Unformatted text preview: − √ 3), let y = arctan( − √ 3), then tan y = − √ 3 for − π 2 < y < π 2 . Therfore y must be the value − π 3 so that arctan( − √ 3) = − π 3 . Because of the limitations on the respective domains of the inverse trigonometric func-tions, arcsin(sin x ) does not necessarily equal x , arctan(tan x ) does not necessarily equal x , etc. . . However, sin(arcsin x ) = x , tan(arctan x ) = x , etc. . . 1. To ±nd arctan(sin 3 π 4 ), ±rst note 3 π 4 is outside the range of values for y = arcsin x . We have sin 3 π 4 = √ 2 2 . Let y = arcsin(sin 3 π 4 ) = arcsin( √ 2 2 ), or equivalently sin y = √ 2 2 . Therefore y = π 4 and arcsin(sin 3 π 4 ) = π 4 . 2. cos(arccos 1 2 ) = 1 2 since arccos 1 2 = π 3 and cos π 3 = 1 2 . 1...
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