use your knowledge of uc to find the values of
TRANSCRIPT
Use your knowledge of UC to find the values of q
1)
Why are there two answers for each?
Section 7-6 The Inverse Trigonometric Functions
Objective: To find values of the inverse trigonometric functions.
Reminder about reference angles
REVIEW CONCEPTS:
Please make sure to have graphs of , and in front of you when lecture starts.
Include the domain and range of each graph.
review
One to one functionRestricted domainInverse functions
–Inverse function equations algebraically.
–Inverse functions graphically.
Trig FUNCTIONS
Sine, cosine and tangent are all functions.
Are they all one-to-one functions?
}n2
x|x{:domain
Notice how the axes are scaled!
Trig functions
Does tan θ have an inverse that is a function? Justify your answer.
If you answered no, you are right. However, we need to have a way to “undo” tan θ.
The graph does not have an inverse function.
}n2
x|x{:domain
Inverse of trig functions
Restrict domain to f(x)=Tan x
Inverse function is Tan-1 x
Notice T is capitalized
Tan x has an inverse.
}2
x2
|x{domain
Notice how the axes are scaled!
Notice how the axes are scaled!
How to Remember which Trigonometric Function is Positive
-1
-1
1
1
S A
T C
AllJust Sine
Just Tangent
Just Cosine
ALLSTUDENTS
TAKE CALCULUS
Sec 7.6 Warm Up Day 2
1) TRUE OR FALSE: and are the same.
2) TRUE OR FALSE: If then
3) Fill in the blank: If then
Notice how the axes are scaled!
Inverse of trig functions
Restrict domain to:
F(x)=Sin x
Inverse function is Sin-1 x
-3π/2 -π -π/2 π/2 π 3π/2
-3
-2
-1
1
2
3
x
y
f(x)= sin xF(x)= Sin x
Notice how the axes are scaled!
-2 -1 1 2
-π/2
π/2
x
y
Notice the axes!
Inverse of trig functions
Restrict domain to 0 <x < F(x)=Cos x
Inverse function is Cos-1 x
-3π/2 -π -π/2 π/2 π 3π/2
-3
-2
-1
1
2
3
x
y
f(x)= cos xF(x)= Cos x
Notice how the axes are scaled!
-2 -1 1 2
-3π/2
-π
-π/2
π/2
π
3π/2
x
y
Notice the axes!
graphs of trig functions
graph of inverse funtions
Restricted domains summarized:
}9090{
}22
{
Tan
OR
Tan
}1800{
}0{
Cos
OR
Cos
}9090{
}22
{
Sin
OR
Sin
Notice the restricted domain for each of the inverse functions includes the first quadrant… i.e. the reference angle.
Sine Function:
sin(30°) = 0.5
Inverse Sine: Sin-1(0.5) = 30°
tan(angle value)= ratioTan-1(ratio)= angle value
sin(angle value)= ratioSin-1(ratio)= angle value
cos(angle value)= ratioCos-1(ratio)= angle value
Example 1
Find Tan-1 2 with a calculator. 0<θ<360°
First make sure your calculator is in the correct mode.
Example 2
Find Tan-1 (-1) without a calculator. 0<θ<2
Restricted domains summarized:
}9090{
}22
{
Tan
OR
Tan
}1800{
}0{
Cos
OR
Cos}9090{
}22
{
Sin
OR
Sin
Use your knowledge UC and RA:
Given
Fill in the blanks below.
Given fill in the blanks below.
Warm UP Day 2::
Graph (in degrees or radians):
Identify the four quadrants on each graph.
Parent graph: sine function
x
y
-1
0
1
sin θ
2
2
3 2
θradians
Q1 Q2 Q3 Q4
review graphing:
Study the graphs below. Draw inverse of each graph.
Example 3
A.) Find Sin-1 (-0.8) with a calculator. 0<θ<360°
B.) Find Cos-1 (-0.5) without a calculator. 0<θ<2
Example 4Hint: pay attention to restricted domain.
)3
2Tancos( 1
A) Find the value of the above expression without a calculator.
Example 4Hint: pay attention to restricted domain.
)3
2Tancos( 1 B) Find the value of
with a calculator.
No matter the mode setting is…
Homework
Sec 7.6 written exercisesDay 1: #1-9 All, read #10Day 2: #11-21 All
Review
http://hotmath.com/help/gt/miniprecalcgt/section_5_1.html