# Download Section 7-6 The Inverse Trigonometric Functions

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```Warm Up
1) Solve: sin π =
1
2
for 0° β€ π β€ 360° 30 °, 150 °
2) Solve: cos π = β
3) If π πππ =
4) If cosπ =
7
β
11
1
β
2
2
2
πππ 0 β€ π β€ 2π
ππ ππ
,
π π
find πππ‘π in exact form.
ππ
π π
β
=β
π
π
find ππ ππ in exact form.
π π
π
5) What is the πππ‘π ÷ ππ ππ ÷ π πππ? ππππ π½
Section 7-6 The Inverse
Trigonometric Functions
Objective: To find values
of the inverse
trigonometric functions
Trig FUNCTIONS
ο¬Sine, cosine and tangent are
all functions
ο¬Are they all one-to-one
functions?
Domain: {x | x οΉ
ο°
2
ο« nο° }
Does the graph
have an inverse?
No!
Notice how the axes are
scaled!
Domain: {x | x οΉ
How can you restrict
the domain to make the
graph one-to-one?
ο°
2
ο« nο° }
π π₯ = ππππ₯
Tan x has an
inverse.
Notice
T is capitalized
Notice how the axes are
scaled!
ο°
ο°
Domain: {x | ο­ οΌ x οΌ }
2
2
Notice how the axes are
scaled!
Notice how the axes are
scaled!
How can you
restrict the domain
to make the graph
one-to-one?
π
π
π₯ β β€π₯β€
2
2
Restrict domain to:
F(x)= Sin x
f(x)= sin x
y
3
2
1
x
-3Ο/2
-Ο
-Ο/2
Ο/2
-1
-2
-3
Ο
3Ο/2
Inverse function is Sin-1 x
y
Ο/2
x
-2
1
-1
2
-Ο/2
Notice how the axes are
scaled!
π π₯ = πππ π₯
How can you restrict the domain
to make the graph one-to-one?
Restrict domain to 0 <x < ο°
y
F(x)= Cos x
f(x)= cos x
3
2
1
x
-3Ο/2
-Ο
-Ο/2
Ο/2
-1
-2
-3
Ο
3Ο/2
π­βπ π = πͺππβπ π
y
3Ο/2
Ο
Ο/2
x
-2
-1
1
-Ο/2
-Ο
-3Ο/2
2
ο­ο°
ο°
Sin ο± ο½ {ο±
ο£ο± ο£ }
2
2
Q1 & Q4
OR
Sin ο± ο½ {ο± ο­ 90ο― ο£ ο± ο£ 90ο― }
πΆππ π₯
ππππ₯
Tanο± ο½ {ο± ο­
OR
ο°
ο°
Cosο± ο½ {ο± 0 ο£ ο± ο£ ο° }
οΌο± οΌ }
2
2
Q1 & Q4
Tanο± ο½ {ο± ο­ 90 οΌ ο± οΌ 90 }
ο―
ο―
ππππ₯
OR
Q1 & Q2
Cosο± ο½ {ο± 0ο° ο£ ο± ο£ 180ο°}
Inverse Trig Functions
ο¬Remember, finding the inverse is
finding an angle!
π ππ
β1
1
= 30°
2
Because: sin 30 ° =
1
2
Example 1
ο¬Find Tan-1 2 in radians with a calculator.
ο¬First make sure your calculator is in the correct
mode.
Example 2 Find Tan-1 (-1) without a calculator.
Find Tanο­1 ο¨ ο­1ο© without a calculator.
"The number whose tangent is ο­ 1"
Domain of Tanx is
sin x
tan x ο½
ο½ ο­1
cos x
3ο° 7ο°
οxο½
,
4 4
οxο½ο­
ο°
4
π
β
2
<π₯<
π
2
Evaluate in radians without a
calculator.
1. πΆππ
β1
β
3
2
cosπ₯ = β
3
2
and 0 β€ π₯ β€ π
πΆππ
2. πππ
β1
β1
sinπ₯ = β1 and
β1
π
β
2
πππβ1
3. πππ
β1
3
tanπ₯ = 3 and
3
5π
β
=
2
6
β€π₯β€
π
2
π
β1 = β
2
π
β
2
πππβ1
<π₯<
π
2
π
3 =
3
Hint: pay
attention to
restricted domain.
βπ
πππ π»ππ
π
β
π
Find the value of the
above expression
without a calculator.
If the ππππ =
2
β
3
what is the cosΞΈ?
Tan domain is
π
β
2
<π<
π
,
2
which is Quad I and Quad IV
Since the tangent is negative, use Quadrant IV
ο¦
ο­1 ο¦ ο­2 οΆ οΆ
Find cos ο§ Tan ο§ ο· ο· without a calculator.
ο¨ 3 οΈοΈ
ο¨
"Cosine of the number
2
whose tangent is ο­ "
3
2
tan x ο½ ο­
3
3
3 13
ο½
2cos ο± ο½
ο½
9ο«4 ο½ c
13
HYP
13
c ο½ 13
ο±
3
13
ο­2
Example 4
cos( Tan
ο­1
2 B) Find the value of
)
3 with a calculator.
No matter the mode setting isβ¦
Find the value without a
calculator
1. πππ
π
π
βπ π
πͺππ
π
2. πππ πΊππ
βπ
β
π
ππ
ππ
π
Chapter 7 Test Thursday, March 2
No Calculator:
ο¬Find exact values of trig functions
ο¬Solve simple trig equations
ο¬Given one trig function, find the
others
ο¬Know key points on graphs
ο¬Find inverses of Sin, Cos, Tan
Calculator:
ο¬Sector area and arc length
ο¬Reference angles
ο¬Coterminal angles