Precalculus Lesson Plans
Chapter 4C – Inverse Trig Functions
Objective
Date
1/30/2017
1/31/2017
2/1/2017
2/2/2017
Day
Monday
Tuesday
Wednesday
Thursday
2/3/2017
Friday
2/6/2017
Monday
2/7/2017
Tuesday
Review
2/8/2017
Wednesday
Test
#1
Discuss College Project
Modeling Lab
Modeling Lab
Graphs of Inverse Trig
Using Inverse Trig to solve
Equations with Unit circle
Using Inverse Trig to solve
Equations using Calculator
Home Enjoyment
Work on your College Project due 2/20
Work on your College Project
Finish lab
Ch. 4C HW #1, Study for Quiz
Quiz, Ch. 4C HW #2 and CTJ
Ch. 4C HW #3
Homework Quiz Ch. 4C
Ch. 4C HW #4 - Study for Test
Test Ch. 4C, LBB due
Work Factoring practice for Thursday.
Graphs of Inverse Trig Functions
p. 322 (Sec. 4.7) #17, 18, Calculator (radians), 19, 21, 23
A. (1) arcsec 4, (2) arcsec 0.25, (3) arccsc (-3) (4) arccot 5,
(Ans: (1) 1.318, (2) NP, .25 is not in the domain of arcsec which is |x| > 1, (3) -.340, (4) .197)
p. 322 Noncalculator #53, 54, 55, 58, 59, 60, 61
B. Calculus Readiness: Draw a function machine for the following functions and find the
domain and range: (1) arcsin x/3, (2) arccos 2x, (3) tan (arccos x/3)
(Ans: (1) D: -3 < x < 3, R:  − π , π  (2) D: [- ½, ½ ] R:[ 0, π], (3) D:[- 3,0) È (0, 3] R: all reals)



2 2
C. Study for Quiz: Graphing Inverse Trig Functions, domains, ranges
#2 Inverse Trig Functions and the Unit Circle
p. 322 (Sec. 4.7) All Noncalculator: #5-11 all, 103 and 104 (Justify both true and false.)
111, 112, 113, 114
A. Find the exact value using unit circle. (noncalculator)
(1) tan (arccos
(Ans: (1)
1
3π
5π
1
) (2) arcsin (tan −
) (3) arccos (cos
) (4) sec (arccos − )
2
2
4
3
3 (2)
π (3) π (4) -2
2
3
B. Calculus Readiness: Draw a function machine for the following functions and find the domain & range:
(1) y = ln (sin x) (restrict answer to interval [0,2π] (2) y = csc(arccos x/2), (3) y = csc(arcsin x/5)
(Ans: (1) D: (0, π) R: (-∞,1)
(2) D: (-2, 2)] R: [1, ¥)
(3) D: [-5, 0) ∪ (0, 5] R: |y| > 1)
C. Critical Thinking Journal (Ch 4C CTJ = 20 points. Remember, you may not use pronouns in CTJ’s.
This discussion is all about domain and range so be careful whether you are discussing the original
trig functions or the inverse trig functions)
(1) Explain why attempting to find sin-1(1.003) on the calculator creates an error message.
(Don’t just say “not in domain”, explain where that domain came from.)
(2) Solve for a in a2 = 9 and a = 9 . Explain the difference.
(3) Solve (exact answer) for “a” in ½ = cos a and cos−1 ½ = a. Explain the difference in solving for “a”.


(4) Since arcsine is the inverse of sine, why is arcsin  sin
5π
6
5π
 π
.
 = and not
6
 6
PreCalculus - Ch. 4C Inverse Trig
#3
2
Solving Conditional Equations with Inverses on the Calculator
Solve the following conditional equations with your calculator over the specified domain.
C. Domain – All degrees
A. Domain – [0, 2π]
B. Domain - All reals
(10) sin x = −0.867
(1) sin x = 0.123
(4) sin x + 4 = 4.56
(2) cos x = 0. 456
(5) 2 cos x = 0.678
(11) cos x = −0.209
(6) tan x −5 = 2.34
(12) tan x = 1.56
(3) tan x = −6.500
(13) sec x = 8.95
(7) sec x = −6.85
(14) csc x = 4.5
(8) csc x = −3.91
(9) cot x = 2.89
(15) cot x = −2.45
Answers:
(1)
{0.123, 3.018}
(2)
{1.097, 5.186}
(3)
{1.723, 4.865}
(9)
x = .333 + kπ, kJ
(10)
x = 240.112 + 360k, x = 299.888 + 360k, kJ
(4)
x = .594 + 2kπ, x = 2.547 + 2kπ, kJ
(11)
x = 102.064 + 360k, x = 257.936 + 360k, kJ
(5)
x= 1.225 + 2kπ, x = -1.225 + 2kπ, kJ
(12)
x = 57.339 + 180k, kJ
(6)
x = 1.435 + kπ, kJ
(13)
x = 83.585 + 360k, x = −83.585 + 360k, kJ
(7)
x = 1.717 + 2kπ, x = 4.566 + 2kπ, kJ
(14)
x = 12.840 + 360k, x = 167.160 + 360k, kJ
(8)
x = 3.400 + 2kπ, x = -.259 + 2kπ, kJ
(15)
x = −22.203 + 180k, kJ
#4 Test tomorrow. Be able to work all the problems on the review sheet. Half test with
calculator and half without. Hand in Little Blue Book.
Factoring Practice:
A. Factor the following in your notebook: (You need this practice for next chapter.)
1)
2)
3)
4)
5)
x2 + 7x
x2 + 6x + 8
x2 + 13x + 30
x2 + 10x + 25
x2 - 8x + 15
6)
x2 - 6x + 9
7)
x2 - 10x + 9
8)
x2 - 3x - 18
9)
x2 - x - 30
10) x2 - x - 2
2
17) 4a2 - 8a - 21
18) 3x2 + x - 2
11) x - x - 42
19) 3x2 - 4x + 1
12) y2- 4y - 12
20) 8y2 - 22 + 5
13) 2a2 - 9a - 5
21) 2tan2 x - 5 tan x
14) 3c2 - 8c + 4
22) sin2 x + 7sin x + 6
23) 2cos2 x - 3cos x + 1
15) 2x2 + 6x
16) 6y2 - 11y - 10
Don’t forget to work on your College Project due 2/20/2017
24) 2sin x cos x + cosx
Precalculus HW Ch. 4C Inverse Trig
3
Review
I. Be able to match the domain, range, and graphs of all six inverse trig functions. Be able
to draw function machine to find domain and ranges as in HW #2.
II. Find the exact value from the unit circle. (noncalculator)
−2
(8) csc−1 (−2)
(5) sin−1 − ½
3
(3) arc csc
(1) arc sin
1
3
(9) cot −1 3
2
(6) cos −1 −
−1
2
(2) arccos 1
(4) tan −1
−−1
(7) sec (−2)
3
III. Find answers on your calculator in radians. Round answers three places behind the
decimal. If no answer write Æ (empty set) and explain why.
(10) sin−1(−.645)
(11) cos−1 (.2135)
(12) arctan (6.54)
(13) arccos (3.5)
(14) cot−1 (8.2)
(15) sec−1 (−7.5)
(16) csc−1 (6.2)
(17) arcsec (−.32)
(18) sin (cos−1 .657)
(19) csc (sec−1 6.5)
(20) sec (arctan 4.5)
(21) tan (arcsec 4.5)
Find the exact value using unit circle. (noncalculator)
1
)
2
3π
(23) arccos (tan −
)
4
(22) tan (arcsin
5π
)
3
1
(25) csc (arcsin − )
2
(24) arcsin (sin
V. Conditional Equations: Solve for x (Noncalculator when I ask for exact.)
(26) sin2 x = ½ (Exact [0, 2π])
− 2
(27) cos x =
(exact, [0°, 360°])
2
(28) tan x = − 3 (Exact [0, 2π])
(29) sin x = 0.456 (decimal, all reals)
Answers Review:
(1)
π/3, (2) 0,
(3) −π/3, (4) −π/6,
(5) −π/6, (6) 3π/4, (7) 2π/3, (8) −π/6
(9) π/6
(10) −.701
(11) 1.356
(12) 1.419
(13) Æ, 3.5 is not in the domain of
(14)
(15)
(16)
(17)
Arccos which is [−1, 1]
.121
1.705
.162
Æ, −.32 is not in the domain of
arcsec which is |x| > 1
(18) .754
(19) 1.012
(20) 4.610
(30) tan x = 3.567 (decimal, all reals)
(31) cos x = −0.123 (decimal, all degrees)
(32) csc x = 0.452 (decimal, all degrees)
(33) cos x = .567 (decimal, [0, 2π])
(34) cot x = - 1.345 (decimal, [0°, 360°])
(21) 4.387
(22) 1
3
(23)
(24)
(25)
(26)
0
−π/3
−2
π 3π 5π 7π
4
,
4
,
4
,
4
(27) 135°, 225°
(28) 2π , 5π
3
3
(29) x = 0.473 + 2kπ, x=2.668+2kπ, k∈J
(30) x = 1.297 + kπ, k∈ J
(31) x = 97.065° + 360k, x=262.935°+360k, k∈J
(32) Æ
(33) .968, 5.315
(34) 143.369°, 323.369
Little Blue Book of PreCalculus Properties
Ch. 4C – Inverse Trig Functions
4C-1
4C-2
4C-3
4C-4
4C-5
4C-6
4C-7
4C-8
4C-9
Inverse Trig Functions (Why is domain of the trig function restricted when creating
the inverse trig function? What quadrants do you look at for the answers for inverse
sin, inverse cos, and inverse tan trig functions? What is the difference in arcsin 0.8
and solving the equation sin x = 0.8? (Answer question and find answers on calculator
in radian mode))
ƒ(x) = arcsin x (definition, two notations, graph, domain, range, how to find
arcsin 1 on calculator – must give answer in radians)
6
ƒ(x) = arccos x (definition, two notations, graph, domain, range, how to find
arccos 1 on calculator – must give answer in radians)
6
ƒ(x) = arctan x (definition, two notations, graph, domain, range, asymptotes, how to
find arctan 1 on calculator – must give answer in radians)
6
ƒ(x) = arccsc x (definition, two notations, graph, domain, range, asymptotes, write the
formula for finding arccsc 6 on calculator – must give answer in radians)
ƒ(x) = arcsec x (definition, two notations, graph, domain, range, asymptotes, write the
formula for finding arcsec 6 on calculator – must give answer in radians)
ƒ(x) = arccot x (definition, two notations, graph, domain, range, asymptotes, write the
formula for finding arccot 6 on calculator – must give answer in radians)
Domain of Composite functions involving Inverse trig functions (explain how to use
the “function machine” to find the domain of tan(arccos x ). Answer in radian mode)
3
Solving Conditional Equations by hand: (How to solve conditional equations by hand.
Use the steps to solve sin x = -½ on [0, 2p], what is the difference in the answers if
domain is all reals, give answers to both)
4C-10 Solving Conditional Equations using Inverse on Calculator: (3 steps for solving
conditional equations using the inverse function of your calculator. Use the steps to
solve sin x = -0.7, [0, 2π], also solve with domain of all reals)