Transcript Chapter 5: Trigonometric Functions

Chapter 5: Trigonometric
Functions
Lessons 3, 5, 6: Inverse Cosine, Inverse
Sine, and Inverse Tangent functions
Mrs. Parziale
Graph y = cos (x).
• Notice that since it fails the
HLT, its inverse is not a
function.
• Restrict it so that the section
contains the entire range (-1
to 1) and passes the HLT.
y
2
–6.28
–3.14
3.14
–2
6.28
x
When restricting the domain, the
following qualifications must be met:
• 1. Must include the values from 0
degrees to 90 degrees (to
represent all acute angles that are
possible in a right triangle.)
• 2. Must include the entire range
of the cosine graph from -1 to 1.
• 3. Make the function continuous
(no breaks), if possible.
y
2
–6.28
–3.14
3.14
–2
6.28
x
Plot the Inverse Function
• Find the domain and range of each.
y
y
3.14
2
–2
–6.28
–3.14
3.14
6.28
2
x
x
–3.14
–2
f ( x)  cos x
• Domain:
• Range:
f ( x)  cos1 x  Arc cos x
Domain:
Range:
Example 1 & 2:
3
• Evaluate cos ( )
2
1
(exact answer)
1
-1
1
cos
(.1256) , give an answer in
• Evaluate
degrees (approximate)
Graph y = sin (x).
• Notice it fails the HLT, so the inverse is not a
function.
• Restrict it so that the section contains the
entire range
(-1 to 1), and passes HLT.
y
2
1
–6.28
–3.14
3.14
–1
–2
6.28 x
When restricting the domain, the
following qualifications must be met:
• 1. Must include the values from 0
degrees to 90 degrees (to
represent all acute angles that are
possible in a right triangle.)
• 2. Must include the entire range
of the sine graph from -1 to 1.
• 3. Make the function continuous
(no breaks), if possible.
y
2
1
–6.28
–3.14
3.14
–1
–2
6.28 x
Plot the Inverse Function
• Find the domain and range of each.
y
y
2
3.14
1
–6.28
–3.14
3.14
6.28 x
–2
2
x
–1
–3.14
–2
f ( x)  sin x
• Domain:
• Range:
f ( x)  sin 1 x  Arc sin x
Domain:
Range:
Examples 1, 2, & 3
• Find the exact value of
1 1
sin
2
• Find the exact value of
 3
sin 

 2 
1
• Find the exact value of Arcsin 1
1
-1
Graph y = tan (x).
• Does this function pass the
HLT?
• How do we restrict the
tangent function so that the
inverse is also a function
and the entire range is
contained (as the domain)
in the new function?
y
2
1
–3.14
3.14 x
–1
–2
When restricting the domain, the
following qualifications must be met:
• 1. Must include the values from 0
degrees to 90 degrees (to
represent all acute angles that are
possible in a right triangle.)
• 2. Must include the entire range
of the tangent graph. (all reals)
• 3. Make the function continuous
(no breaks), if possible.
y
2
1
–3.14
3.14 x
–1
–2
Plot the Inverse Function
• Find the domain and range of each.
y
y
2
3.14
1
–3.14
3.14 x
–2
2
x
–1
–3.14
–2
f ( x)  tan x
• Domain:
• Range:
f ( x)  tan 1 x  Arc tan x
Domain:
Range:
Examples 1 & 2
• Find the exact value of
3
1
Arctan
3
• Find the exact value of
 3
tan 

 3 
1
1
-1
Closure
• Name three conditions that must be true
when restricting the domain of the trig
functions to graph the inverse functions?
• Looking at the unit circle, what quadrants is
cosine restricted to? Sine? Tangent?
• Try these
cos
1
3
 2
sin 

 2 
1