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Inverse Trig Functions 6.1

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Inverse Trig Functions6.1

JMerrill, 2007

Revised 2009

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

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

- 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

- Quadrant IV is
- Quadrant I is
- Answers must be in one of those two quadrants or
the answer

doesn’t exist.

- How do we draw inverse functions?
- Switch the x’s and y’s!

Switching the x’s and y’s also means switching the axis!

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

- y = arcsin x or y = sin-1 x
- Both mean the same thing. They mean that you’re looking for the angle (y)where sin y = x.

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

- 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.

- 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

- We must restrict the domain
- Now the inverse

- Quadrant I is
- Quadrant II is
- Answers must be in one of those two quadrants or
the answer

doesn’t exist.

- We must restrict the domain
- Now the inverse

–1.5

1.5

–

2

–1.5

1.5

–

–3

3

–

Graphing Utility:Graph the following inverse functions.

Set calculator to radian mode.

a. y = arcsin x

b. y = arccos x

c. y = arctan x

Graphing Utility:Approximate the value of each expression.

Set calculator to radian mode.

a. cos–1 0.75

b. arcsin 0.19

c. arctan 1.32

d. arcsin 2.5

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

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

- 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!

- Find the exact value of
- Steps:
- Draw a triangle using only the info inside the parentheses.
- Now use your x, y, r’s to answer the outside term

3

2

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

12

-7

- Find the exact value of