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(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(x) (x,y) (y, x)

Sometimes you just dont 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 xs and ys y = arcsin (x) solve for y f(x) = arcsin (x) write using function notation f(x) = sin(x)

Inverse Trigonometric function notation: Inverse sine

Inverse cosine

Inverse tangent

FINDING INVERSE OF A TRIG FUNCTION ALGEBRAICALLY :

Given y = sin (x) - = sin ( -/6) sin (-/6) = - (-/6) = sin (- ) switch x and y values (-/6) = arcsin ( - ) solve for y

(-/6) = sin( - ) sin( - ) = (-/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: 2 Y-int.(0,0)

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

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

Xy-values of sin xy = arcsin xx values of sin x

Graphs of inverse functions The graph of y = arc sin x Domain: Range:

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ARCCOSINE Sketch the common curve of y = cos xSelect interval of the cosine that will be one-to-oneSketch the graph of arccosine

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.

Graphs of inverse functions The graph of y = arccos x Domain: Range:

The other trig functions require similar restrictions on their domains in order to generate an inverse.y=tan(x)y=arctan(x)

Graphs of inverse functions The graph of y = arctan x Domain: Range:

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 ( /3) = therefore

Evaluating Inverse Trigonometric Functions algebraically.

Find the exact value for

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

Cos ( 3/4) = ; therefore