Inverse Trigonometric

Functions4.7

How do you determine if a function has an inverse? It must be one to one … pass the

horizontal line test Will a sine, cosine, or tangent

function have an inverse?

Their inverses are defined over the following intervals:

Sine: [ -π/2, π/2 ] Cosine: [ 0, π ] Tangent: [ -π/2, π/2 ]

Notation for inverse trig functions:

y = sin-1x or y = arcsin x i.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 them

Chart from page 324

Let’s evaluate inverse trig functions:

1.) 2.) 2

2arccos23arcsin

It will be helpful to

remember:

sinθ = y then arcsin y = θ

cos θ = x then arccos x

= θ

Try some on your own…

1.)

2.)

3.)

21arcsin

2arccos

1arctan

Calculator….

To do this on your calculator…▪ 2nd then trig function

▪ The steps are on page 325 if you need a refresher

▪Let’s practice….▪Pg. 328 #’s 4 – 28 (4’s)

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) = x arcsin(sin y) =

y cos(arccos x) = x arccos(cos y) =

y tan(arctan x) = x arctan(tan y) =

y *remember - only works over certain

intervals…▪ Refer to page 326

What should you do if it lies outside the range?

Use the co-terminal angles that are in the range!

Let’s practice: 1) arcsin[sin (π/2) ] = ?

2) arccos[cos (π/6) ] = ?

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

4) arcsin[sin (3π/4)]= ?

A few more….

5) arcsin(sin 5π/3 ) = ? 6) sin(arcsin π ) = ?

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

Pg. 328 #’s 44, 46, 48

Day 1 HW….

Pg. 328 #’s 3 – 63 by 3’s

5 pt pass opportunity…

Find the exact value of:

sin ¿

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 triangle

Some more practice….

2.) sin(arccos √5/ 5) 3.) csc[arctan(-5/12)]

Writing algebraic expressions… Ex.1) sin(arccos 3x) ; 0 ≤ x ≤ 1/3

Ex 2) cot(arcsin 2x) ; 0 ≤ x ≤ 1/3

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

HW DAY 2…..Pg. 328

45-69 (3’s), 71, 73, 91, 93, 95