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9/24/2012 MTH 113 Section 4.7 Inverse Trig Functions Review: Only one to one functions have inverses; they must pass the Only one‐to‐one functions have inverses; they must pass the horizontal line test as well as the vertical line test. f ( x) = x 2 f ( x) = x 2 , x ≥ 0 f‐1 is the symbol for the inverse of f. f‐1 does not mean the reciprocal of f. If the point (a,b) is on the graph of f, then (b,a) is on the graph of f‐1. f and f‐1 are reflections of each other about the line y = x. f and f are reflections of each other about the line y = x 1 9/24/2012 2 9/24/2012 A Function and Its Inverse 3 9/24/2012 Find the exact value of sin −1 0. Find i d the h exact value l off sin i −1 − 1. 1 Find the exact value of sin −1 − 3 . 2 4 9/24/2012 Find the exact value of cos−1 − 1. Find the exact value of cos −1 2 . 2 Find the exact value of cos −1 3 . 2 5 9/24/2012 6 9/24/2012 7 9/24/2012 Find the exact value of tan −1 3 . 3 Find the exact value of tan −1 − 1. Find the exact value of tan −1 − 3. Using a Calculator to Evaluate Inverse Trigonometric Functions 8 9/24/2012 Use a calculator to find each of the following in radians rounded to four decimal places. sin‐1 0.47 tan‐1(‐30) sin‐1 (‐0.625) tan −1 − 5061 ( ) cos‐1 (4/9) ⎛ 7⎞ cos −1 ⎜⎜ ⎟⎟ ⎝ 10 ⎠ Inverse Properties The Sine Function and its Inverse sin(sin‐1x) = x for every x in [‐1, 1] x) = x for every x in [‐1 1] ‐1 sin (sinx) = x for every x in ⎡⎢− π , π ⎤⎥ ⎣ 2 2⎦ The Cosine Function and its Inverse cos(cos‐1x) x) = x for every x in [ x for every x in [‐1, 1, 1] 1] ‐1 cos (cos x) = x for every x in [0, π] 9 9/24/2012 The Tangent Function and its Inverse tan(tan‐1x) = x for every real number x tan‐1(tan x) = x for every x in ⎛ − π , π ⎞ ⎜ ⎝ 2 ⎟ 2⎠ Find exact values without the use of a calculator. cos(cos‐10.57) 2π ⎞ ⎛ cos −1 ⎜ cos ⎟ 3 ⎠ ⎝ 4π ⎞ ⎛ cos −1 ⎜ cos ⎟ 3 ⎠ ⎝ 4π 4π is not in [ 0, π ] , so is not the answer. 3 3 You must find what angle in [0, π ] has the same cosine as that of 4π . 3 10 9/24/2012 tan(tan‐1 380) ⎡ ⎛ π⎞ ⎤ tan −1 ⎢ tan ⎜ - ⎟ ⎥ ⎣⎢ ⎝ 3 ⎠ ⎥⎦ ⎡ 3π ⎤ tan −1 ⎢ tan ⎥ 4 ⎦ ⎣ cos‐1(cos 2π) cos(cos‐13π) 11 9/24/2012 Use a sketch to find exact values. sin(tan‐1 7/24) cos[sin‐1(‐4/5) tan[cos‐1(‐ ¼)] ⎡ ⎛ 2 ⎞⎤ sec ⎢sin -1 ⎜⎜ − ⎟⎟ ⎥ ⎢⎣ ⎝ 2 ⎠ ⎥⎦ 12