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