6.6 – TRIG INVERSES AND THEIR GRAPHS

Pre-Calc

Inverse Trig Functions

Original Function

Inverse

y = sin x y = sin-1 x y = arcsin x

y = cos x y = cos-1 x y = arccos x

y = tan x y = tan-1 x y = arctan x

REVIEW SLIDE

Consider the graph of y = sin x

What is the domain and range of sin x?

What would the graph of y = arcsin x look like?

What is the domain and range of arcsin x?

Domain: all real numbersRange: [-1, 1]

Domain: [-1, 1]Range: all real numbers

REVIEW SLIDE

Is the inverse of sin x a function? This will also be true for

cosine and tangent. Therefore all of the

domains are restricted in order for the inverses to be functions.

REVIEW SLIDE

How do you know if the domain is restricted for the original functions?

Capital letters are used to distinguish when the function’s domain is restricted.

Original Functions with

Restricted Domain

Inverse Function

y = Sin x y = Sin-1 x y = Arcsin x

y = Cos x y = Cos-1 x y = Arccos x

y = Tan x y = Tan-1 x y = Arctan x

REVIEW SLIDE

Original Domains Restricted Domains

Domain Range

y = sin x

all real numbers

y = Sin x y = sin x y = Sin x

y = cos xall real

numbers

y = Cos x y = cos x y = Cos x

y = tan xall real

numbers except n,

where n is an odd integer

y = Tan x y = tan x

all real numbers

y = Tan x

all real numbers

REVIEW SLIDE

x

yy = sin(x)

12

2

3

3

2

2

4

2

1

6

002

1

6

2

2

4

2

3

3

12

)(

xfx

Sketch a graph of y = Sin x Remember principal values

Create a table

21

32

342

262

10062

142

232

32

1

)(sin 1

xx

Now use your table to generate Sin-1

12

2

3

3

2

2

4

2

1

6

002

1

6

2

2

4

2

3

3

12

)sin(

xx

IF YOU CAN REMEMBER AND MEMORIZE WHAT THE original and inverse funcitons look like, it will make your life a lot easier!!!

Table of Values of Cos x and Arccos xy = Cos x

X Y

0 1

π/3 0.5

π/2 0

2π/3 -0.5

π -1

y = Arccos x

X Y

1 0

0.5 π/3

0 π/2

-0.5 2π/3

-1 π

x

y

x

y

The other trig functions require similar restrictions on their domains in order to generate an inverse.

Like the sine function, the domain of the section of the

tangent that generates the arctan is .

2,

2

,

2,

2RandD

2,

2,

RandD

Y=Tan(x)Y=Arctan(x)

Table of Values of Tan x and Arctan xy = Tan x

X Y

-π/2 UD

-π/4 -1

0 0

π/4 1

π/2 UD

y = Arctan x

X Y

UD -π/2

-1 -π/4

0 0

1 π/4

UD π/2

Write an equation for the inverse of y = Arctan ½x. Then graph the function and its inverse.

To write the equation:1.Exchange x and y2.Solve for y

x = Arctan ½yTan x = ½y2Tan x = y

Let’s graph 2Tan x = y first.Complete the table:

Then graph!

y = 2Tan x

X Y

-π/2 Undefined

-π/4 -2

0 0

π/4 2

π/2 Undefined

Now graph the original function, y = Arctan ½x by switching the table you just completed!

1

-1

π/2-π/2

Now graph the original function, y = Arctan ½x by switching the table you just completed!

y = 2Tan x

X Y

-π/2 UD

-π/4 -2

0 0

π/4 2

π/2 UD

y = Arctan ½x

X Y

UD -π/2

-2 -π/4

0 0

2 π/4

UD π/2 1-1

π/2

-π/2

Write an equation for the inverse of y = Sin(2x).

Then graph the function and its inverse.

To write the equation:1.Exchange x and y2.Solve for y

x = Sin(2y)Arcsin(x) = 2y½Arcsin(x) = y

Let’s graph y = Sin(2x) first.The domain changes because of the 2, how?

Now graph the inverse function, y = Arcsin(x)/2 by switching the table you just completed!

y = Sin2x

X Y

-π/4 -1

-π/12 -.5

0 0

π/12 .5

π/4 1

1

-1

π/2 π-π/2

y = ½Arcsin(x)

X Y

-1 -π/4

-.5 -π/12

0 0

.5 π/12

1 π/4Remember you can always check and see if they are symmetric with respect to y = x

€

−2

≤ 2x ≤π

2

€

−4

≤ x ≤π

4

Divide all sides by 2

Graph the inverse of:

€

y =π

2+ Arc sin x

Let’s find the inverse equation first:

€

x =π

2+ Arc sin y

Flip the “x” and “y” and solve for “y”:

€

x −π

2= Arc sin y

€

Sin x −π

2

⎛

⎝ ⎜

⎞

⎠ ⎟= y

X Y

0 -1

π/4

π/2 0

3π/4

π 1

€

y = Sin x −π

2

⎛

⎝ ⎜

⎞

⎠ ⎟

€

− 22

€

22

Take the sine of both sides

Domain is now:

€

−2

≤ x −π

2≤π

2

€

0 ≤ y ≤ π

Add π/2 to all three sides

1

-1

π/2 π

Graph

€

y =π

2+ Arc sin x

y = (π/2)+Arcsin x

X Y

-1 0

π/4

0 π/2

3π/4

1 π

Since we are graphing Arcsin the domain will become the range, but it will change!!Solve for x:

€

y =π

2+ Arc sin x

€

y −π

2= Arc sin x

€

sin y −π

2

⎛

⎝ ⎜

⎞

⎠ ⎟= x

Domain is now:

€

−2

≤ y −π

2≤π

2

€

0 ≤ y ≤ π

Add π/2 to all three sides

Take the sine of both sides

€

− 22

€

22

€

−2

≤ x ≤π

2

What is the domain for Sin(x)?

1-1

π/2

π

Now make a table using the y-values as your input into this function:

y = Arcsinx

Just shifted up π/2

Now try to graph just by using the shifting technique.

€

y = Arc sin x −π

4

1-1

π/2

π

y = Arcsinx

Just shifted down π/4

-π/2

Now try to graph just by using the shifting technique.

€

y = Arc tan x +π

4

1-1

π/2

π

y = Arctan(x)

Just shifted up π/4

-π/2

Graph:

€

y = Arc tan x −π

2

Determine if each of the following is true or false. If false give a counter example.

1. Cos-1(cos x) = x for all values of x

2. Sin-1(sin x) = x for all values of x

3. Sin-1x + Cos-1x = π/2 -1 ≤ x ≤ 1

4. Arccos x = Arccos (-x) -1 ≤ x ≤ 1

5. Tan-1x = 1/(Tan x)

x = 270°

Cos-1(cos 270°) = Cos-1(0) = 90°FALSE

FALSE

TRUE

FALSE

FALSE

x = 180° or try x = 225°

Sin-1(sin 180°) = Sin-1(0) = 0°

x = 0 of try x = 1 or -1

Sin-1(0) + Cos-1(0) = 0° + 90°= 90°

x = 1 or try x = -1

Arccos(1) ≠ Arccos (-1) 0° ≠ 180°

x = 0

Tan-1 (0) ≠ 1 / (Tan (0)) 0° ≠ 1 / 0 UNDEFINDED

Evaluate each expression

-30 degrees 45 degrees

Evaluate each expression

1

Negative square root of 3