INVERSE vs. RECIPROCAL

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Inverse Trig Functions
Inverse Trig Solve for the Angle
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INVERSE vs. RECIPROCAL

• Its important not to confuse an INVERSE trig
  function with a RECIPROCAL trig function.
• The reciprocal of SINE is COSECANT
  (sin x ) -1 csc x
• The inverse of SINE is ARCSIN
  sin -1 x arcsin x
• The notation is very important - be careful !!

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Evaluating INVERSE Trigs

• Evaluating an inverse trig means finding the
  ANGLE that gives us the stated ratio.For
  example, to evaluate arcsin 0.5, (read the
  arcsine of 0.5) we must ask ourselves for what
  angle does sin 0.5 ?

Answer 30 or p/6 radians
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How Many Answers are Right?

• There are other angles for which the sine of that
  angle 0.5. For example
• sin 30 0.5
• sin 150 0.5
• sin 390 0.5
• The way we handle this is to give a general
  solution, that is written in a way such that all
  possible solutions are included in it.

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General Solutions

• Your solution can include all angles that are
  co-terminal to the solution angle by adding
  360k or 2pk to the angle, where k is any
  integer (positive or negative).In our example,
  the first obvious angle was 30 or p/6
  radians arcsin 0.5
  30 360k

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General Solutions

• To be complete, the solution must include angles
  from other quadrants that produce the same sign
  and value. Think of this as the reverse of the
  process we used when we found reference angles
  --- this time we know the reference angle is 30
  and the sign is positive, so we have to find the
  angle in quadrant II ( the other quadrant where
  sine is positive ) that shares the same
  reference angle ( 30 ). The answer ? . . .

150
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General Solutions

• Our complete, general solution is
• arcsin 0.5 30 360k 150 360kOR, in
  radians
• arcsin 0.5 p/6 2pk 5p/6 2pk
• Solve these
• arccos ( -0.5 )
• arcsec 1

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General Solutions

• Solutions
• arccos ( -0.5 ) 120 360k 240 360k
• OR arccos ( -0.5 ) 2p/3 2pk 4p/3
  2pk
• arcsec 1 0 360kOR arcsec 1 0 2pk
• Notice that in our second example there is only
  one angle between 0 and 360 where secant 1.
  The secant of 180 -1.

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The Briefest Answer

• Evaluate arctan ( -1 )
• First find the Quad I angle where tan 145 or
  p/4 radians.
• Now, in which quadrants is tan negative
  ?Quadrants II and IV
• Angles 135 360k 315 360kBut this
  answer can be simplified to 135 180k. Why does
  this satisfy ALL the angles ?

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Practice Problems

• arcsin ( -1 )
• arccos ( v2/2 )
• arctan ( -v3 )
• arccsc v2
• arccot 0
• arcsec ( -2 )

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Summary

• When evaluating a trig function for a given
  angle, there is ONE ANSWER. Remember all the
  rules for evaluating trig functions reference
  coterminal angles, positive negative quadrants,
  etc.
• When evaluating an inverse trig function, there
  can be INFINITE ANSWERS.

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Summary

• First find the angle in quadrant I that yields
  the desired ratio ( the number given in the
  inverse trig function ).
• Then determine which quadrant ( s ) the solution
  is in to achieve the given sign - sometimes there
  is only 1 angle, that is OK.
• State the angle answer ( s ) for those quadrants.

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Summary

• Add a multiple of k to include all co-terminal
  solutions ( i.e. 360k or 2pk ) for each of
  your angle solutions.
• Finally, it may be possible to make your answer
  more brief, like we did with arctan. This is
  usually the case when the solutions differ by
  exactly 180 or p radians.