Download Section 4.6 Inverse Trigonometric fuctions

Objectives:
-Evaluate and Graph inverse trig functions
-Find compositions of trig functions

Is the sine function one-to-one?


No! Does not pass HLT.
If we restrict the domain of the sine function to
𝜋 𝜋
the interval [− , ], the function IS one-to-one.
2 2

The inverse equation is y = sin -1x and is
graphed by reflecting the restricted y = sinx in
the line y = x.

Notice that the domain of the inverse is [-1, 1]
𝜋 𝜋
and its range is [− , ].
2 2

Because angles and arcs given on the unit circle
have equivalent radian measures, the inverse
sine function is sometimes referred to as the
arcsine function y = arcsin x.


sin -1 x or y = arcsin x can be interpreted as the angle (or arc)
𝜋
𝜋
between − 𝑎𝑛𝑑 with a sine of x.
2
2
-1
For example, sin 0.5 is the angle with a sine of 0.5.



(30 degrees)
Recall that sin t is the y-coordinate of the point on the unit circle
that corresponds to the angle or arc length, t.
Because the range of the inverse sine function are restricted, the
possible angle measures of the inverse sine function are located
on the right half of the unit circle
A)
−1 2
sin
2
B) arcsin −
C) sin−1 −2
D) sin−1 0
3
2



To make the cosine function one-to-one, the
domain must be restricted to [0, ].
The inverse cosine function is y = cos -1x or
arccosine function y = arccos x.
The graph of y = cos -1 x is found by reflecting the
graph of the restricted y = cosx in the line y = x.


Recall that cos t is the x-coordinate of the point on
the unit circle that corresponds to the angle or arc
length, t.
Because the range of y = cos -1x is restricted to [0,
] , the possible angle measures of the inverse
cosine function are located on the upper half of
the unit circle
A) cos−1 1
B) arccos −
3
2
C) cos −1 3
D) cos −1 −1

To make the tangent function one-to-one, the
𝜋 𝜋
domain must be restricted to (− , ).
2 2


The inverse tangent function is y = tan-1x or
arctangent function y = arctan x.
The graph of y = tan-1 x is found by reflecting the
graph of the restricted y = tanx in the line y = x.
Unlike sine and cosine, the domain of the
inverse tangent function is (-, )



On the unit circle, tant =
sin 𝑥
cos 𝑥
or tant =
𝑦
𝑥
The values of y = tan-1x will be located on the right
𝜋
𝜋
half of the unit circle, not including − and
2
2
because the tangent function is undefined at those
points.
A)
−1 3
tan
3
B) arctan 1
C) arctan (− 3)

Rewrite the function in one of the following
forms:
sin y = x
 cos y = x
 tan y = x



Make a table of values, assigning radians from
the restricted range to y.
Plot the points and connect them with a
smooth curve.
A) y =
𝑥
arctan
2
y
x
B) y = sin−1 2𝑥
y
x
C) y =
𝑥
arccos
4
y
x
A) In a movie theater, a 32-foot-tall screen is located 8 feet
above ground. Write a function modeling the viewing
angle θ for a person in the theater whose eye-level when
sitting is 6 feet above ground.
B) Determine the distance that corresponds
to the maximum viewing angle.



In Lesson 1.7, you learned that if x is in the domain
of f(x) and f -1(x) then
f [f -1(x)] = x and f -1[f(x)] = x
Because the domains of the trig functions are
restricted to obtain the inverse trig function, the
properties do not apply for all values of x.
For example, while sin x is defined for all x, the
domain of sin-1 x is [-1,1].


Therefore, sin(sin-1 x) = x is only true when -1 ≤ x ≤ 1.
A different restriction applies for the composition
𝜋 𝜋
of sin-1(sinx) because the domain of sin x is [− , ].

Therefore,
sin-1(sinx)
= x is only true when
𝝅
−
𝟐
2 2
≤𝒙≤
𝝅
𝟐
A) sin
1
arcsin
2
B)
C) arctan tan
cos −1
5𝜋
cos
2
5𝜋
−
2
D) arcsin
7𝜋
sin
6
A) sin
4
−1
cos
5
B) cos
C) tan
5
arccos −
13
8
arcsin
17



arccosecant (sec -1)
arcsecant
(csc -1)
arccotangent (cot -1)
EXTRA EXAMPLE: Find the exact value.
A) csc
3
−1
cot
4
C) sec
15
−1
cos −
17
B) tan
13
−1
csc
5
D) cot
sec −1
25
−
7
A) Write cot (arccos x) as an algebraic expression
of x that does not involve trigonometric functions.
B) Write cos(arctan x) as an algebraic expression of
x that does not involve trigonometric functions.
1. sin  tan 1

2.
3.
7 

24 
 1 
3 
csc cos  
 
2


 
 1  3  
sin  tan    
 4 
