8D.1: Inverse Trig. Functions

Name:
Date:
Period:
8D.1: Inverse Trig. Functions
Angles from Ratios
Suppose we are building a roof truss like the
one pictured and we want to figure out the
angle to cut the boards. That is, we need to
find the value of # in the diagram.
Can you find the value of # in degrees?
6E?.
#
20E?.
By definition, the trigonometric functions give us a ratio for a given angle. If we want to find an
angle from a given ratio, we must use the “inverse” function. In the example above, we need to use
+
the “inverse tangent” of ,- which is written in the following ways:
6
6
6
# = tan/0 1 5 ,
67 # = arctan 1 5 ,
67 # = atan 1 5
20
20
20
where the first two are the most standard ways.
+
When we read tan/0 9,-:, we interpret it as,
"<ℎ>? @A ?ℎB CB>AD7B 6E >F >FGHB ?ℎ>? ℎ>A > ?>FGBF? 6E 6/20? "
Inverse Trigonometric Ratios as Functions
We now want to define inverse trigonometric ratios as functions. Keep in mind that when we have
a function, we are required to maintain the characteristic that
"E67 BLB7M N L>HDB @F ?ℎB O6C>@F, ?ℎB7B @A PQRS PQT M L>HDB @F ?ℎB 7>FGB. "
Also, remember that the inverse of a function reverses the domain and range of the original
function.
The inverse trig functions we will consider are
M = sin/0 N ,
M = cos/0 N ,
M = tan/0 N ,
What’s the Domain and Range?
67 M = arcsin N
67 M = arccos N
67 M = arctan N
Consider the definition of an inverse function and fill in the domains and ranges below
M = sin N
Domain
Range
M = sin/0 N
M = cos N
M = cos/0 N
M = tan N
M = tan/0 N
Graphing Inverse Functions
Try These Evaluate on the appropriate range without your calculator. (Think of the unit circle)
a. sin/0(−1) =
b. sin/0 9− ,: =
c. sin/0(0) =
d. sin/0 9− : =
0
0
,
e. sin/0(1) =
Graph It!
Now use the answers above to help you graph M = sin/0 N
on the interval [−1,1].
Try These Evaluate on the appropriate range without your calculator. (Think of the unit circle)
a. cos/0(−1) =
b. cos/0 9− ,: =
c. cos/0(0) =
d. cos/0 9− ,: =
0
0
e. cos/0(1) =
Graph It!
Now use the answers above to help you graph M = cos/0 N
on the interval [0, d].
Try These Evaluate on the appropriate range. Don’t use your calculator for the first 3.
a. tan/0 (−1) =
b. tan/0 (0) =
c. tan/0 (1) =
d. tan/0 (−2) =
e. tan/0 (2) =
Graph It!
Now use the answers above to help you graph M = tan/0 N
on the interval [−5,5].
2|Pa ge
Revised:3/5/2014
Putting it all together
Find the exact value without using your calculator
0
a. sin 9e6A /0 ,:
b. tan(sin/0(1))
√,
c. tan 9arcsin 9 , ::
g
d. arcsin 9cos 9 + ::
3|Pa ge
Revised:3/5/2014

Download Report