Download Section 7-6 The Inverse Trigonometric Functions

Warm Up
1) Solve: sin 𝜃 =
1
2
for 0° ≤ 𝜃 ≤ 360° 30 °, 150 °
2) Solve: cos 𝜃 = −
3) If 𝑠𝑖𝑛𝜃 =
4) If cos𝜃 =
7
−
11
1
−
2
2
2
𝑓𝑜𝑟 0 ≤ 𝜃 ≤ 2𝜋
𝟑𝝅 𝟓𝝅
,
𝟒 𝟒
find 𝑐𝑜𝑡𝜃 in exact form.
Quad IV
𝟕𝟐
𝟔 𝟐
−
=−
𝟕
𝟕
find 𝑐𝑠𝑐𝜃 in exact form.
Quad II
𝟐 𝟑
𝟑
5) What is the 𝑐𝑜𝑡𝜃 ÷ 𝑐𝑠𝑐𝜃 ÷ 𝑠𝑒𝑐𝜃? 𝒄𝒐𝒔𝟐 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 

2
 n }
Does the graph
have an inverse?
No!
Notice how the axes are
scaled!
Domain: {x | x 
How can you restrict
the domain to make the
graph one-to-one?

2
 n }
𝑓 𝑥 = 𝑇𝑎𝑛𝑥
Tan x has an
inverse.
Notice
T is capitalized
Notice how the axes are
scaled!


Domain: {x |   x  }
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
𝑥 − ≤𝑥≤
2
2
Restrict domain to:
F(x)= Sin x
f(x)= sin x
y
3
2
1
x
-3π/2
-π
-π/2
π/2
-1
-2
-3
π
3π/2
Inverse function is Sin-1 x
y
π/2
x
-2
1
-1
2
-π/2
Notice how the axes are
scaled!
𝑓 𝑥 = 𝑐𝑜𝑠𝑥
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
-3π/2
-π
-π/2
π/2
-1
-2
-3
π
3π/2
𝑭−𝟏 𝒙 = 𝑪𝒐𝒔−𝟏 𝒙
y
3π/2
π
π/2
x
-2
-1
1
-π/2
-π
-3π/2
2


Sin   {
  }
2
2
Q1 & Q4
OR
Sin   {  90    90 }
𝐶𝑜𝑠𝑥
𝑇𝑎𝑛𝑥
Tan  { 
OR


Cos  { 0     }
  }
2
2
Q1 & Q4
Tan  {  90    90 }


𝑆𝑖𝑛𝑥
OR
Q1 & Q2
Cos  { 0    180}
Inverse Trig Functions
Remember, finding the inverse is
finding an angle!
𝑠𝑖𝑛
−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 Tan1  1 without a calculator.
"The number whose tangent is  1"
Domain of Tanx is
sin x
tan x 
 1
cos x
3 7
x
,
4 4
x

4
𝜋
−
2
<𝑥<
𝜋
2
Evaluate in radians without a
calculator.
1. 𝐶𝑜𝑠
−1
−
3
2
cos𝑥 = −
3
2
and 0 ≤ 𝑥 ≤ 𝜋
𝐶𝑜𝑠
2. 𝑆𝑖𝑛
−1
−1
sin𝑥 = −1 and
−1
𝜋
−
2
𝑆𝑖𝑛−1
3. 𝑇𝑎𝑛
−1
3
tan𝑥 = 3 and
3
5𝜋
−
=
2
6
≤𝑥≤
𝜋
2
𝜋
−1 = −
2
𝜋
−
2
𝑇𝑎𝑛−1
<𝑥<
𝜋
2
𝜋
3 =
3
Hint: pay
attention to
restricted domain.
−𝟏
𝒄𝒐𝒔 𝑻𝒂𝒏
𝟐
−
𝟑
Find the value of the
above expression
without a calculator.
If the 𝑇𝑎𝑛𝜃 =
2
−
3
what is the cosθ?
Tan domain is
𝜋
−
2
<𝜃<
𝜋
,
2
which is Quad I and Quad IV
Since the tangent is negative, use Quadrant IV

1  2  
Find cos  Tan    without a calculator.
 3 

"Cosine of the number
2
whose tangent is  "
3
2
tan x  
3
3
ADJ
3 13

2cos  

94  c
13
HYP
13
c  13

3
13
2
Example 4
cos( Tan
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
𝟒
𝟑
−𝟏 𝟕
𝑪𝒐𝒔
𝟒
2. 𝒔𝒆𝒄 𝑺𝒊𝒏
−𝟏
−
𝟑
𝟏𝟑
𝟏𝟑
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