Download Section 7-6 The Inverse Trigonometric Functions

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Warm Up
1) Solve: sin πœƒ =
1
2
for 0° ≀ πœƒ ≀ 360° 30 °, 150 °
2) Solve: cos πœƒ = βˆ’
3) If π‘ π‘–π‘›πœƒ =
4) If cosπœƒ =
7
βˆ’
11
1
βˆ’
2
2
2
π‘“π‘œπ‘Ÿ 0 ≀ πœƒ ≀ 2πœ‹
πŸ‘π… πŸ“π…
,
πŸ’ πŸ’
find π‘π‘œπ‘‘πœƒ in exact form.
Quad IV
πŸ•πŸ
πŸ” 𝟐
βˆ’
=βˆ’
πŸ•
πŸ•
find π‘π‘ π‘πœƒ in exact form.
Quad II
𝟐 πŸ‘
πŸ‘
5) What is the π‘π‘œπ‘‘πœƒ ÷ π‘π‘ π‘πœƒ ÷ π‘ π‘’π‘πœƒ? π’„π’π’”πŸ 𝜽
Section 7-6 The Inverse
Trigonometric Functions
Objective: To find values
of the inverse
trigonometric functions
Trig FUNCTIONS
Sine, cosine and tangent are
all functions
Are they all one-to-one
functions?
Domain: {x | x ο‚Ή

2
 n }
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?

2
 n }
𝑓 π‘₯ = π‘‡π‘Žπ‘›π‘₯
Tan x has an
inverse.
Notice
T is capitalized
Notice how the axes are
scaled!


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œ‹
πœ‹
π‘₯ βˆ’ ≀π‘₯≀
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!
𝑓 π‘₯ = π‘π‘œπ‘ π‘₯
How can you restrict the domain
to make the graph one-to-one?
Restrict domain to 0 <x < 
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‘­βˆ’πŸ 𝒙 = π‘ͺπ’π’”βˆ’πŸ 𝒙
y
3Ο€/2
Ο€
Ο€/2
x
-2
-1
1
-Ο€/2
-Ο€
-3Ο€/2
2


Sin  ο€½ {
 ο‚£ }
2
2
Q1 & Q4
OR
Sin  ο€½ { ο€­ 90 ο‚£  ο‚£ 90 }
πΆπ‘œπ‘ π‘₯
π‘‡π‘Žπ‘›π‘₯
Tan ο€½ { ο€­
OR


Cos ο€½ { 0 ο‚£  ο‚£  }
 ο€Ό }
2
2
Q1 & Q4
Tan ο€½ { ο€­ 90 ο€Ό  ο€Ό 90 }


𝑆𝑖𝑛π‘₯
OR
Q1 & Q2
Cos ο€½ { 0ο‚° ο‚£  ο‚£ 180ο‚°}
Inverse Trig Functions
Remember, finding the inverse is
finding an angle!
𝑠𝑖𝑛
βˆ’1
1
= 30°
2
Because: sin 30 ° =
1
2
Example 1
Find Tan-1 2 in radians with a calculator.
First make sure your calculator is in the correct
mode.
Example 2 Find Tan-1 (-1) without a calculator.
Find Tanο€­1  ο€­1 without a calculator.
"The number
angle whose tangent is ο€­ 1"
Domain of Tanx is
sin x
tan x ο€½
ο€½ ο€­1
cos x
3 7
οƒžxο€½
,
4 4
οƒžxο€½ο€­

4
πœ‹
βˆ’
2
<π‘₯<
πœ‹
2
Q4
*If the angle is in
Q4, write it as a
negative angle.
Evaluate in radians without a
calculator.
1. πΆπ‘œπ‘ 
βˆ’1
βˆ’
3
2
cosπ‘₯ = βˆ’
πΆπ‘œπ‘ 
2. 𝑆𝑖𝑛
βˆ’1
βˆ’1
3
2
βˆ’1
and 𝑄2
3
5πœ‹
βˆ’
=
2
6
sinπ‘₯ = βˆ’1 and 𝑄4
π‘†π‘–π‘›βˆ’1
3. π‘‡π‘Žπ‘›βˆ’1 3
πœ‹
βˆ’1 = βˆ’
2
tanπ‘₯ = 3 and 𝑄1
π‘‡π‘Žπ‘›βˆ’1
πœ‹
3 =
3
*If the
angle is in
Q4, write it
as a
negative
angle.
Hint: pay
attention to
restricted domain.
βˆ’πŸ
𝒄𝒐𝒔 𝑻𝒂𝒏
𝟐
βˆ’
πŸ‘
Find the value of the
above expression
without a calculator.
If the π‘‡π‘Žπ‘›πœƒ =
2
βˆ’
3
what is the cosΞΈ?
Tan domain is
πœ‹
βˆ’
2
<πœƒ<
πœ‹
,
2
which is Quad I and Quad IV
Since the tangent is negative, use Quadrant IV

ο€­1  ο€­2 οƒΆ οƒΆ
Find cos  Tan  οƒ· οƒ· without a calculator.
 3 οƒΈοƒΈ

"Cosine of the number
2
whose tangent is ο€­ "
3
2
tan x ο€½ ο€­
3
3
ADJ
3 13
ο€½
2cos  ο€½
ο€½
94 ο€½ c
13
HYP
13
c ο€½ 13

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 is…
Find the value without a
calculator
1. 𝒄𝒔𝒄
πŸ’
πŸ‘
βˆ’πŸ πŸ•
π‘ͺ𝒐𝒔
πŸ’
2. 𝒔𝒆𝒄 π‘Ίπ’Šπ’
βˆ’πŸ
βˆ’
πŸ‘
πŸπŸ‘
πŸπŸ‘
𝟐
Chapter 7 Test Thursday, March 2
No Calculator:
Find exact values of trig functions
Solve simple trig equations
Given one trig function, find the
others
Know key points on graphs
Find inverses of Sin, Cos, Tan
Calculator:
Sector area and arc length
Reference angles
Coterminal angles
Convert degrees/radians
Apparent size
Find trig values from (x,y) point
Homework
Page 289 #1-13 odd
Chapter Test
Page 293 #1-12
No calculator
on #6 - 12