Aim: How do we define the inverse of y = sin x as y = Arc sin x?

Do Now: Given f(x) = sin x, a) fill in the table below:

b) write the coordinates of each point after reflected in y = x

2 2

2

2

32x 0

f(x)

2

3

HW: p.424 # 15,21,22,28,29,32,33,37

The coordinates of each point after reflected in y = x are

),2

,1(),,0(),2

3,1(),2,0(

)2,0(),2

3,1(),,0(),

2,1(),0,0(

These points are the inverse of y = sin x

If we connect those points, the graph is called

xy 1sin y = arc sin x or

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-2.0 -1.5 -1.0 -0.5 0.5 1.0 1.5 2.0

-2.0

-1.5

-1.0

-0.5

0.5

1.0

1.5

2.0

x

y

y = x

y = sin x

y = arc sin x

xy 1sin or

x x

y y 1 1

y x sin 1

x x 1 1

y y

For y = sin x, the Domain = { Real numbers}

}

Domain = { }

real numbers}

Range = {

Range = {

Is y x sin 1 a function ?

Generally NO (It fails the vertical line test)

But if we limit the domain then it can be a function over that particular range.

}22

{

yy

y x sin 1

If the domain is

the relation IS a function.

We use y = Arc sin x or to represent the inverse that is a function.

xSiny 1

That is, is only limited in quadrants I and IV only

xSiny 1

Therefore, y = arcsin x is a function only within -½π ≤ x ≤ ½π

We use y = Arcsin x or

y = Sin-1 x to represent

Finally, the inverse function of y = sin x only defined in quadrant I and IV

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-6

-4

-2

2

4

6

x

y

2

130sin 30

2

1sin 1 or 150

Which one is true?

302

11Sin ,since 30 is in quadrant I

#

#sin

The idea between the function and inverse function is

#

#1Sin

2

1sin

Sin 1 1

2

is equivalent to

2

2sin

2

21Sin 45

330Notice that

2

1sin

Sin 1 1

2

2

21Sin is equivalent to2

2sin

1.If Sin 1

2, write the equivalent function

11Sin 2. If in degrees., find the value of

Sin 1 0 3. If , find the value of in radians.

Sin 1 1( )

1.If

, find the value of in radians.4. If

Sin 3

25. If , find the value of

in radians.

6. Find the exact value of if the angle is a third-quadrant angle.

7. Find the exact value of

)5

3sec(arcsin

4

5

)0cos( 1Sin

8. Find the exact value of )17

8cos( 1Sin

9. Find θ to the nearest degree )55.0(sin 1