Transcript Inverse Trig Functions

Inverse Trig
Functions
Principal
Solutions
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°.
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.
But Which Solution?

If you are evaluating the inverse
trig function of a positive
number, it probably won’t
surprise you that the principal
solution is the Quadrant I angle:
Arctan 1 = 45° or π/4 radians
Sin-1 0.5 = 30° or π/6 radians
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?
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
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.
Arcsin/Arccsc

Choose adjacent quadrants with
positive & negative y-values :
Q3
Q4
-π/2
Q1
+
+
π/2
Q3
Q2
π
Q4
3π/2
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 ?
Arcsin/Arccsc

Principal Solutions to Arcsin
must be between -90° and 90° or
- π/2 and π/2 radians, that
includes Quadrant IV angles if
the number is negative and
Quadrant I angles if the number
is positive.
Arccos/Arcsec

Choose adjacent quadrants with
positive & negative y-values :
+
Q4
Q3
-π/2
+
Q1
Q2
π/2
Q3
π
Q4
3π/2
Which quadrant of angles is adjacent to Q1, but with
negative y-values? What range of solutions is valid?
Arccos/Arcsec

Principal Solutions to Arccos
must be between 0° and 180° or
0 and π radians, that includes
Quadrant II angles if the number
is negative and Quadrant I
angles if the number is positive.
Arctan/Arccot

Choose adjacent quadrants with
positive & negative y-values :
Q3
-π
Q4
-π/2
Q2
Q1
π/2
π
Which quadrant of
angles is adjacent
to Q1, over a
continuous section,
but with negative yvalues? What range
of solutions is
valid?
Arctan/Arccot

Principal Solutions to Arctan
must be between -90° and 90° or
-π/2 and π/2 radians, that
includes Quadrant IV angles if
the number is negative and
Quadrant I angles if the number
is positive.
Practice
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Arcsin (-0.5)
Arctan 0
Arcsec 2
Arccot √3
Arccos (-1)
Arccsc (-1)
Summary - Part 1

If the inverse trig function begins with a
CAPITAL letter, find the one, principal
solution.
Arcsin & Arccsc: -90° to 90° / -π/2 to π/2
Arccos & Arcsec: 0° to 180° / 0 to π
Arctan & Arccot: -90° to 90° / -π/2 to π/2
Compound Expressions #1

 3 
 Evaluate: sin Arc tan 3  Arc cos

 2 

(Start inside the parentheses.)
 3 
sin Arc tan 3  Arc cos 
 2 

 sin 60  150
 sin( 210)
1

2
Compound Expressions #2

7 

Evaluate. arcsin sin

6 
 1 
 arcsin  
2
7
11

 2k;
 2k
6
6
NOTE: We cannot forget to include all relevant
solutions and all of their co-terminal angles.
Practice
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tan Arc tan( 1)
5 

Arc cos tan
 4 
 
 3 
arcsin cosArcsin  

2





