Inverse Trig Functions 6'1

1
Inverse Trig Functions6.1

• JMerrill, 2007
• Revised 2009

2
Recall

• From College Algebra, we know that for a function
  to have an inverse that is a function, it must be
  one-to-oneit must pass the Horizontal Line Test.

3
Sine Wave

• From looking at a sine wave, it is obvious that
  it does not pass the Horizontal Line Test.

4
Sine Wave

• In order to pass the Horizontal Line Test (so
  that sin x has an inverse that is a function), we
  must restrict the domain.
• We restrict it
• to

5
Sine Wave

• Quadrant IV is
• Quadrant I is
• Answers must be in one of those two quadrants or
• the answer
• doesnt exist.

6
Sine Wave

• How do we draw inverse functions?
• Switch the xs and ys!

Switching the xs and ys also means switching
the axis!
7
Sine Wave

• Domain/range of restricted wave?
• Domain/range of inverse?

8
Inverse Notation

• y arcsin x or y sin-1 x
• Both mean the same thing. They mean that youre
  looking for the angle (y) where sin y x.

9
Evaluating Inverse Functions

• Find the exact value of
• Arcsin ½
• This means at what angle is the sin ½ ?
• p/6
• 5p/6 has the same answer, but falls in QIII, so
  it is not correct.

10
Calculator

• When looking for an inverse answer on the
  calculator, use the 2nd key first, then hit sin,
  cos, or tan.
• When looking for an angle always hit the 2nd key
  first.
• Last example Degree mode, 2nd, sin, .5 30.

11
Evaluating Inverse Functions

• Find the value of
• sin-1 2
• This means at what angle is the sin 2 ?
• What does your calculator read? Why?
• 2 falls outside the range of a sine wave and
  outside the domain of the inverse sine wave

12
Cosine Wave
13
Cosine Wave

• We must restrict the domain
• Now the inverse

14
Cosine Wave

• Quadrant I is
• Quadrant II is
• Answers must be in one of those two quadrants or
• the answer
• doesnt exist.

15
Tangent Wave
16
Tangent Wave

• We must restrict the domain
• Now the inverse

17
Graphing Utility Graphs of Inverse Functions
Graphing Utility Graph the following inverse
functions.
Set calculator to radian mode.
a. y arcsin x
b. y arccos x
c. y arctan x
18
Graphing Utility Inverse Functions
Graphing Utility Approximate the value of each
expression.
Set calculator to radian mode.
a. cos1 0.75
b. arcsin 0.19
c. arctan 1.32
d. arcsin 2.5
19
Composition of Functions

• Find the exact value of
• Where is the sine
• Replace the parenthesis in the original problem
  with that answer
• Now solve

20
Example

• Find the exact value of
• The sine angles must be in QI or QIV, so we must
  use the reference angle

21
Example

• Find tan(arctan(-5))
• -5
• Find
• If the words are the same and the inverse
  function is inside the parenthesis, the answer is
  already given!

22
Example

• Find the exact value of
• Steps
• Draw a triangle using only the info inside the
  parentheses.
• Now use your x, y, rs to
  answer the outside term

3
2
23
Last Example

• Find the exact value of
• Cos is negative in QII and III, but the inverse
  is restricted to QII.

12
-7
24
You Do

• Find the exact value of