Inverse Trig Functions

1
Inverse Trig Functions

• Principal Solutions

2
Principal Solutions

• In the last section we saw that an INVERSE TRIG
  function has infinite solutions
• arctan 1 45 180k
• But there is only one PRINCIPAL SOLUTION, 45.

3
Principal Solutions

• Each inverse trig function has one set of
  Principal Solutions. (If you use a calculator to
  evaluate an inverse trig function you will get
  the principal solution.)
• We give the Principal Solution when the inverse
  trig function is capitalized, Arcsin or Sin-1.

4
But Which Solution?

• If you are evaluating the inverse trig function
  of a positive number, it probably wont surprise
  you that the principal solution is the Quadrant I
  angle
• Arctan 1 45 or p/4 radians
• Sin-1 0.5 30 or p/6 radians

5
Negative Numbers?

• But if you are evaluating the inverse trig
  function of a negative number, you must decide
  which quadrant to use.
• For Arcsin Arccsc Q3 or Q4?
• For Arccos Arcsec Q2 or Q3?
• For Arctan Arccot Q2 or Q4?

6
The Right Choice

• There is a clear set of rules regarding which
  quadrants we choose for principal inverse trig
  solutions
• For Arcsin Arccsc use Q4
• For Arccos Arcsec use Q2
• For Arctan Arccot use Q4

7
But WHY?

• The choice of quadrants for principal solutions
  was not made without reason. The choice was made
  based on the graph of the trig function. The
  next 3 slides show the justification for each
  choice.

8
Arcsin/Arccsc

• Choose adjacent quadrants with positive
  negative y-values

Q3 and 4 are not adjacent to Q1, unless we look
to the left of the y-axis. Which angles in Q4 are
adjacent to Q1 ?
9
Arcsin/Arccsc

• Principal Solutions to Arcsin must be between
  -90 and 90 or - p/2 and p/2 radians, that
  includes Quadrant IV angles if the number is
  negative and Quadrant I angles if the number is
  positive.

10
Arccos/Arcsec

• Choose adjacent quadrants with positive
  negative y-values

Which quadrant of angles is adjacent to Q1, but
with negative y-values? What range of solutions
is valid?
11
Arccos/Arcsec

• Principal Solutions to Arccos must be between 0
  and 180 or 0 and p radians, that includes
  Quadrant II angles if the number is negative and
  Quadrant I angles if the number is positive.

12
Arctan/Arccot

• Choose adjacent quadrants with positive
  negative y-values

Which quadrant of angles is adjacent to Q1, over
a continuous section, but with negative y-values?
What range of solutions is valid?
13
Arctan/Arccot

• Principal Solutions to Arctan must be between
  -90 and 90 or -p/2 and p/2 radians, that
  includes Quadrant IV angles if the number is
  negative and Quadrant I angles if the number is
  positive.

14
Practice

• Arcsin (-0.5)
• Arctan 0
• Arcsec 2
• Arccot v3
• Arccos (-1)
• Arccsc (-1)

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Summary - Part 1

• If the inverse trig function begins with a
  CAPITAL letter, find the one, principal solution.
• Arcsin Arccsc -90 to 90 / -p/2 to p/2
• Arccos Arcsec 0 to 180 / 0 to p
• Arctan Arccot -90 to 90 / -p/2 to p/2

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Compound Expressions 1

• Evaluate
• (Start inside the parentheses.)

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Compound Expressions 2

• Evaluate.

NOTE We cannot forget to include all relevant
solutions and all of their co-terminal angles.
18
Practice