inv_trig_fns - M 408 K Fall 2005 Inverse Trig Functions...

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M 408 K Fall 2005 Inverse Trig Functions Important Decimal Approximations and Useful Trig Identities Decimal Approximations: 0.000 0 = 12 0 0.000 0 p = = 0.500 2 1 = 12 2 0.524 6 = = ( ) ÷ ø ö ç è æ = = = 2 3 2 1 3 1 0.577 3 3 12 3 0.785 4 = = 0.707 2 2 = ; 0.866 2 3 = 12 4 1.047 3 = = 1.000 1 = 12 6 1.571 2 = = 2 3 1 1.155 3 3 2 = = 12 8 2.094 3 2 = = 2 2 1 1.414 2 = = 12 9 2.356 4 3 = = ( ) 2 1 2 3 1 3 1.732 3 ÷ ø ö ç è æ = = = 12 10 2.618 6 5 = = Trigonometric Identities: adj a hyp opp c b u SOH-CAH-TOA sin (u) = opp / hyp tan (u) = opp / adj sec (u) = hyp / adj cos (u) = adj / hyp v u When u + v = pi / 2 = 90 degrees, sin (u) = b / c = cos (v) = cos ( pi/2 - u ) , tan (u) = b / a = cot (v) = cot ( pi/2 - u ) , sec (u) = c / a = csc (v) = csc ( pi/2 - u ) . sin ( - x ) = - sin ( x ) tan ( - x ) = - tan ( x ) cos ( - x ) = cos ( x ) sec ( - x ) = sec ( x )
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Inverse Sine Function: u = sin -1 (x) , x Î [ -1 , 1 ] , u ú û ù ê ë é - Î 2 , 2 p . u = sin -1 (x) when sin (u) = x and u ú û ù ê ë é - Î 2 , 2 . u -1 1 2 -2 1 2 1/2 .707 .866 x -1 -2 -.866 -.707 -1/2 Locate x on the vertical axis. sin (u) = x u = For x in [ -1 , +1 ] , u = inverse sine (x) When x is in [ 0 , 1 ] , u = inv sin (x) is in [ 0, pi/2 ] When x is in [ -1 , 0 ] , u = inv sin (x) is in [ - pi/2, 0 ] x = 0.77 sin -1 x) ( = 0.882 radians u -1 1 1 1/2 .707 .866 x -1 -.866 -.707 -1/2 sin (u) = x u = u = inverse sine (x) x = -0.95 sin -1 x) ( = -1.244 radians u -1 1 1 1/2 .707 .866 x -1 -.866 -.707 -1/2 sin (u) = x u = u = inverse sine (x) x = -0.31 sin -1 x) ( = -0.311 radians
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Table of values: y = sin -1 (x) x y = sin 1 (x) ------------------------- --------------------- - 1 - π / 2 = - 1.571 - √3 / 2 = -0.866 - π / 3 = - 1.047 - √2 / 2 = -0.707 - π / 4 = - 0.785 -1/2 - π / 6 = - 0.524 0 0 1/2 π / 6 = 0.524 √2 / 2 = 0.707 π / 4 = 0.785 √3 / 2 = 0.866 π / 3 = 1.047 1 π / 2 = 1.571 u -1 1 1 1/2 .707 .866 x -1 -.866 -.707 -1/2 sin (u) = x u = u = inverse sine (x) x = 0.61 sin -1 x) ( = 0.652 radians u -1 1 1 1/2 .707 .866 x -1 -.866 -.707 -1/2 sin (u) = x u = u = inverse sine (x) x = 0.97 sin -1 x) ( = 1.339 radians a = Sqrt( 1 - x^2 ) x 1 cos (u) = adj / hyp u Drawing the angle u = inv sin (x) when x > 0 : sin (u) = x = x/1 = opp / hyp , Make opp = x and hyp = 1 . cos ( inv sin (x) ) = Sqrt( 1 - x^2 ) u = inv sin (x) 0 1 -1 .5 pi/2 pi/3 pi/4 pi/6 1 2 -pi/4 -pi/6 -1 x -pi/3 -2 -pi/2 -.5 y y = y = inv sin (x) x = 0.30 sin -1 x) ( = 0.310
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Inverse Tangent Function: u = tan -1 (x) , x Î ( - infinity , infinity ) , u ÷ ø ö ç è æ - Î 2 , 2 p . u = tan -1 (x) when tan (u) = x and u ÷ ø ö ç è æ - Î 2 , 2 . u 1 -1 -2 2 -3 3 1 1 2 -4 4 x 2 3 3 4 -1 -1 -2 -2 -3 -3 -4 Locate x on the number line placed vertically with the origin at point (1,0) . tan (u) = x u = As x -> 0, u = inv tan (x) -> 0 As x -> infinity, u = inv tan (x) -> pi / 2 As x -> - infinity, u = inv tan (x) -> - pi / 2 For x in ( -inf, inf ) , u = inverse tangent (x) x = 3.33 tan -1 x) ( = 1.279 radians
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Table of values: y = tan -1 (x) x y = tan -1 (x) ------------------ ------------------------- -10 - 1.471 -5 - 1.373 - 3 = -1.732 - π /3 = - 1.047 -1 - π /4 = - 0.785 - 3 / 3 = -0.577 - π /6 = - 0.524 0 0 3 / 3 = 0.577 π /6 = 0.524 1 π /4 = 0.785
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This note was uploaded on 05/02/2011 for the course MATH 408D taught by Professor Chu during the Spring '09 term at University of Texas at Austin.

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inv_trig_fns - M 408 K Fall 2005 Inverse Trig Functions...

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