Inverse Trigonometric Functions 4.7

Inverse Trigonometric Functions4.7How do you determine if a function has an inverse?It must be one to one pass the horizontal line testWill a sine, cosine, or tangent function have an inverse?

Their inverses are defined over the following intervals:Sine: [ -/2, /2 ]Cosine: [ 0, ]Tangent: [ -/2, /2 ]

Notation for inverse trig functions:y = sin-1x or y = arcsin xi.o.i sin y = x

y = cos-1x or y = arccos x

y = tan-1x or y = arctan x

Their graphs are on pg 324 if you would like to reference themChart from page 324

Lets evaluate inverse trig functions:1.) 2.)

It will be helpful to remember:

sin = y then arcsin y =

cos = x then arccos x =

Try some on your own1.)

2.)

3.)

Calculator.To do this on your calculator2nd then trig function

The steps are on page 325 if you need a refresher

Lets practice.Pg. 328 #s 4 28 (4s)Homework Day 1.Pg. 324 #s 3 42 (by 3s)

Compositions of Functions

f (f-1 (x) ) = ? f-1 ( f(x) ) = ? Therefore:sin(arcsin x) = xarcsin(sin y) = ycos(arccos x) = xarccos(cos y) = ytan(arctan x) = xarctan(tan y) = y

*remember - only works over certain intervalsRefer to page 326

What should you do if it lies outside the range?Use the co-terminal angles that are in the range!Lets practice:1) arcsin[sin (/2) ] = ?

2) arccos[cos (/6) ] = ?

3) tan[arctan (-5)] = ?

4) arcsin[sin (3/4)]= ?A few more.5) arcsin(sin 5/3 ) = ? 6) sin(arcsin ) = ?

7) arctan (tan /6) = ? 8) tan(arcsin 2/2)=?

Pg. 328 #s 44, 46, 48Day 1 HW.Pg. 328 #s 3 63 by 3s5 pt pass opportunityFind the exact value of:Inverse Trig Day 2!!Using right triangles to find exact values of compositions of inverse functions.

Ex. 1) Find the exact value of tan(arccos 2/3)Use a right triangleSome more practice.2.) sin(arccos 5/ 5) 3.) csc[arctan(-5/12)] Writing algebraic expressionsEx.1) sin(arccos 3x) ; 0 x 1/3

Ex 2) cot(arcsin 2x) ; 0 x 1/3

Practice pg. 328 #s 60, 64, 66, 68Pg. 331 #104 b, d

HW DAY 2..Pg. 328 45-69 (3s), 71, 73, 91, 93, 95