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
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