Lesson 4.7. Inverse Trigonometric Functions.Previously you have learned

To find an inverse of a function, let every x be y and every y be x, then solve the equation for y.

Inverse function notation f¯¹(x)

For a function to have an inverse it has to be one-to-one. One x for one y value, and one y for one x value. It will pass the vertical and the horizontal line test.

Two inverse functions on the graph reflect over y=x

f(x) f¯¹(x) (x,y) (y, x)

Sometimes you just don’t have a nice or convenientalgebraic process that will give you an inverse function.

Many functions need a special, new rule for their inverse. Some examples of these functions are:

FINDING INVERSE OF A TRIG FUNCTION :

Given f(x) = sin (x) y = sin (x) change f(x) for y x = sin (y) switch x’s and y’s

y = arcsin (x) solve for y f¯¹(x) = arcsin (x) write using function

notation f¯¹(x) = sin¯¹(x)

Inverse Trigonometric function notation: Inverse sine

Inverse cosine

Inverse tangent

FINDING INVERSE OF A TRIG FUNCTION ALGEBRAICALLY :

Given y = sin (x) - ½ = sin ( -π/6) sin (-π/6) = - ½

(-π/6) = sin (- ½ ) switch x and y

values (-π/6) = arcsin ( - ½ ) solve for y

(-π/6) = sin¯¹( - ½ ) sin¯¹( - ½ ) = (-π/6)

How does this look on the graph ??

Graphing Inverse Trigonometric functions:

The graph of y = sin x

D: all reals R: [-1,1] Period: 2π Y-int.(0,0)

- this function is not one-to-one (different x values yield the same y)- with domain restricted to [-π/2 ; π/2] y = sin x is one-to one- therefore, we can use this “piece to finds its inverse

Xy-values of sin x

y = arcsin xx – values of sin x

Using the domain restricted to [-π/2 ; π/2] let’s graph y = arcsin x Remember that y = arcsin x is equivalent to sin y = x

Graphs of inverse functions The graph of y = arc sin x

[ 1,1]

,2 2

Domain:

Range:

NEXT ON THE LIST

• ARCCOSINE • Sketch the common curve of y = cos x• Select interval of the cosine that will be one-

to-one• Sketch the graph of arccosine

x

yy = cos(x)

The chosen section for the cosine is in the red frame. This section includes all outputs from –1 to 1 and all inputs in the first and second quadrants.

Since the domain and range for the section are the domain and range for the inverse cosine are

,1,1,0 and .,01,1 and

x

yy = arccos(x)

Graphs of inverse functions The graph of y = arccos x

[ 1,1]Domain:

Range: [0, ]

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

,

2,2

RandD

2,2

,

RandD

y=tan(x)

y=arctan(x)

Graphs of inverse functions The graph of y = arctan x

Domain:

Range:

( , )

,2 2

The table below will summarize the parameters we have so far. Remember, the angle is the input for a trig function and the ratio is the output. For the inverse trig functions the ratio is the input and the angle is the output.

arcsin(x) arccos(x) arctan(x)

Domain

Range

11 x 11 x x

22

x

20

x

22

x

When x<0, y=arccos(x) will be in which quadrant?

When x<0, y=arcsin(x) will be in which quadrant?

When x<0, y=arctan(x) will be in which quadrant?

y<0 in IV y>0 in II

y<0 in IV

Evaluating Inverse Trigonometric Functions algebraically.When evaluating inverse trigonometric functions, you are “looking for the

angle whose (insert: sin, cos, or tan) is x.” Also, keep in mind the domain and range of each function.

Find the exact value for

Solution: For what value of x is sin (x) = ? Sin ( π/3) = therefore

2

3arcsin

2

3

2

3

32

3arcsin

Evaluating Inverse Trigonometric Functions algebraically.Find the exact value for

Solution: For what value of x is cos (x) = ?

Cos ( 3π/4) = ; therefore

2

2

4

3

2

2cos 1

2

2