MA 154 Lesson 16 DelworthMA 154 Lesson 16 DelworthSection 7.6 The Inverse Trigonometric Functions67,631tanππθθ==32,323sinππθθ==47,421cosππθθ==While this is true for equations with the directions. Find all solutions of the equation in the interval [0, 2), today we are going to define the inverse trigonometric functions.Arcsin = sin-1Arccos = cos-1Arctan = tan-1631tan31arctan1π=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−323sin23arcsin1π=⎟⎟⎠⎞⎜⎜⎝⎛=⎟⎟⎠⎞⎜⎜⎝⎛−421cos21arccos1π=⎟⎠⎞⎜⎝⎛=⎟⎠⎞⎜⎝⎛−Why only one answer? Remember you calculators only returned values in certain quadrants when wewere working with inv. functions? arcsin is only defined from 2π− to 2π, which are quadrants I and IV. arctan is also only defined from 2π− to 2π, which are quadrants I and IV. However, arccos is defined from 0 to , which are quadrants I and II.This is reinforced when you use the inv tan, inv cos, or inv sin keys on you calculator.Find the exact value of the expression whenever it is defined.⎟⎠⎞⎜⎝⎛−−21sin1⎟⎟⎠⎞⎜⎜⎝⎛−22cos1( )1tan1−−⎟⎠⎞⎜⎝⎛21arcsin⎟⎟⎠⎞⎜⎜⎝⎛−23arccos( )3arctan⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛22arcsinsin)]1(cos[cos1−( )[ ]5tantan1−1sin-1sin-1tan-1tan-1cos-1cos-1MA 154 Lesson 16 DelworthSection 7.6 The Inverse Trigonometric Functions⎟⎠⎞⎜⎝⎛45sinarcsinπ⎟⎠⎞⎜⎝⎛45cosarccosπ⎟⎠⎞⎜⎝⎛45tanarctanπ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−−23cossin1⎥⎦⎤⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛−31tancos1⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−−21sintan1Find the exact value of the expression whenever it is defined.⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−135arccos2sin⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−54sin2cos1⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛−−409tan2tan1Write the expression as an algebraic expression in x for x > 0.( )x1sincos−( )xarccos2sin2MA 154 Lesson 16 DelworthSection 7.6 The Inverse Trigonometric FunctionsGraph:xy1sin−=xy1cos−=xy1tan−=xy
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