Download is on the inverse

Unit 3 – Exploring Inverse trig. functions
Standards:

F.BF.4 Find inverse functions.

F.BF.4d (+) Produce an invertible function from a non‐invertible function by restricting the domain.

F.TF.6 (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows it’s
inverse to be constructed.

F.TF.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology,
and interpret them in terms of the context.★
Essential Question(s): Why does the calculator only give one answer for an inverse trig function? Aren't there infinite answers? How do
inverse trigonometric functions help us solve equations?
Notes
1.
2.
3.
4.
5.
If the point (x, y) is on the function, the point (y, x) is on the inverse.
Two functions are inverses of one another if and only if f(f-1(x)) = x and f-1(f(x)) = x.
The inverse of a function will reflect over the line y = x.
In order for a graph to be a function, it must pass the vertical line test (VLT).
In order for a function to have an inverse function, it must pass the horizontal line test (HLT).
Example #1: f(x) = x2. Sketch the graph of f(x) and determine if it passes VLT and HLT. Is the inverse of f(x) a
function?
Example #2: Can we restrict f(x) = x2 such that it passes both VLT and HLT. Sketch the new graph and write how the
function would be restricted.
1
Example #3: Graph 𝑓(𝜃) = sin 𝜃 and the line y = .
2
a. How many times do these functions
intersect between -2π and 2π?
b. How is this graph related to finding the
1
solution to 2 = sin θ?
c. If the domain is not limited, how many
1
solutions exist to the equation = sin θ?
2
d. Would this be true for the other trigonometric functions? Explain.
Example #4: f(x) = sin(x)?
a. Complete the table for f(x) = sin(x). Then sketch the graph
of f(x) = sin x and its reflection across the line y = x.
x
−𝜋
−𝜋
2
0
𝜋
2
𝜋
3𝜋
2
sin(x)
b.
What does your graph tell you about the relationship between the graph of
y = sin(x) and y = sin-1(x)?
c.
Is y = sin-1(x) a function?
d.
What is the domain and range of y = sin(x)?
e.
Can we restrict the range of y = sin-1(x) so that it would be a function? What
would it need to be?
f.
Graph the function y = sin-1(x) with the restricted range to the right.
Example #5: f(x) = cos(x)
a. Complete the table, then graph the function f(x) = cos x
and its reflection over the line y = x.
x
cos(x)
−𝜋
−𝜋
2
0
𝜋
2
𝜋
3𝜋
2
b. What does the graph of above tell you about the relationship between the graph of y = cos(x) and y = cos-1(x)?
c. Is y = cos-1(x) a function?
d. What is the domain and range of y = cos(x)?
e. Can we restrict the range of y = cos-1(x) so that it would be a function? What
would it need to be?
f.
Graph the function y = cos-1(x) with the restricted range to the right.
Example #6: f(x) = tan(x)
a. Complete the table, then graph the function f(x) = tan x and its inverse.
x
−𝜋
2
−𝜋
4
tan(x)
0
𝜋
4
𝜋
2
3𝜋
4
𝜋
5𝜋
4
b. What does the graph of above tell you about the relationship between the graph of y = tan(x) and y = tan-1(x)?
c. Is y = tan-1(x) a function?
d. What is the domain and range of y = tan(x)?
e. Can we restrict the range of y = tan-1(x) so that it
would be a function?
h. Graph the function y = tan-1(x) with the restricted range.
In Summary
We use the names sin-1, cos -1, and tan-1 or ArcSin, ArcCos, and ArcTan to represent the inverse of these functions
on the limited domains you explored above. The values in the limited domains of sine, cosine and tangent are
called principal values. (Similar to the principal values of the square root function.) Calculators give principal
values when reporting sin-1, cos -1, and tan-1.
Complete the chart below indicating the domain and range of the given functions.
Function
𝑓(𝜃) = sin−1 𝜃
𝑓(𝜃) = cos−1 𝜃
𝑓(𝜃) = tan−1 𝜃
Domain
Range
1
Note: “arcsin” in the problem arcsin 2 means the angle whose sin is x.
1
Thus you are to find the sin of what is 2. We need to remember that arcsin is only defined in Quadrant I and IV, so we
𝜋
6
1
2
must find an answer in those quadrants only. Since the sin( ) is the answer is
𝜋
6
The inverse functions do not have ranges that include all 4 quadrants. Add a column to your chart that indicates the
quadrants included in the range of the function. This will be important to remember when you are determining
values of the inverse functions.
1.
Use what you know about trigonometric functions and their inverses to evaluate the following expressions. Two examples
are included for you. (Unit circles can also be useful.)
Example 1:
Example 2:
√3
sin (cos-1 1 + tan-1 1) The answer will be a
ArcCos ( )
The answers will be an
2
number, not an angle.
angle.
Simplify parentheses
first.
-1
-1
√3
θ
=
cos
1
θ
=
tan
1
Let θ = Arccos ( ) Ask yourself, what angle has
𝜋
2
θ =0
θ =
√3
4
a cos value of .
2
𝜋
4
sin (0 + )
√3
Cos θ = ( )
2
Using the definition of
𝜋
Arccos.
sin( ) =
4
𝜋
6
5𝜋
) included?
6
θ=( )
Why isn’t (
So,
√3
2
Substitution
√2
2
So,
sin (cos-1 1 + tan-1 1) =
𝜋
6
ArcCos ( ) = ( )
√2
2
Inverse Properties of Trigonometric Functions
𝜋
𝜋
 If -1 < x < 1 and - 2 ≤ 𝑦 ≤ 2 , then sin(arcsin x) = x and arcsin(sin y) = y
 If -1 < x < 1 and 0 ≤ 𝑦 ≤ 𝜋, then cos(arccos x) = x and arccos(cos y) = y
𝜋
2
𝜋
2
 If x is a real number and - < 𝑦 < , then tan(arctan x) = x and arctan(tan y) = y
𝜋 𝜋
Keep in mind, the properties such as arcsin(sin y) = y is not valid for values of y outside the interval [− 2 , 2 ]
Practice
1
a.
𝜃 = cos −1 2
b.
𝜃 = 𝐴𝑟𝑐𝑠𝑖𝑛 0
c.
sin−1 2
d.
cos (tan−1 √3 − sin−1 2)
e.
cos (tan−1
f.
sin (sin−1
1
√3
)
3
√3
)
2
Work Period – Without using a calculator (using only a unit circle), find the value of each expression in radians.
Remember that the inverse trig functions have a restricted range. Make sure your answer falls in the range for that
function.
1.
2.
sin−1 0
sin−1 1
8.
9.
cos−1 0
cos−1 1
3.
sin−1 ( 2 )
10.
cos−1 ( 2 )
4.
sin−1 ( 2 )
11.
cos−1 ( 2 )
5.
sin−1(−1)
12.
6.
sin−1 (−
13.
cos−1 (−
7.
1
√2
1
)
2
√2
sin−1 (− 2 )
14.
15.
16.
tan−1 0
tan−1 1
1
17.
tan−1( √3 )
√2
18.
tan−1 ( 3 )
cos−1 (−1)
19.
tan−1(−1)
1
)
2
√2
cos−1 (− 2 )
20.
tan−1(− √3 )
21.
tan−1 (−
√3
√3
)
3
Homework – Part 2
1.
Complete the table
Function
y = arcsin x
Alternative Notation
Domain
Range
y = arccos x
y = arctan x
√3
𝜋
1.
cos−1 ( 2 )
8.
tan−1 (tan ( 4 ))
2.
arcos (−
√2
)
2
𝜋
cos−1 (cos ( 3 ))
1
arccos(− )
2
𝜋
−1
cos (cos ( 3 ))
√3
sin−1 ( 2 )
−1
9.
sin−1 (cos ( 2 ))
10.
sin (cos −1 (2))
3.
4.
5.
6.
7.
sin(tan
√3)
11.
𝜋
1
√3
2
cos (tan−1 (√3) − sin−1 ( ))
12.
tan (cos −1 (−
13.
cos−1 ( 2 )
14.
sin−1 (−
√2
))
2
√2
5𝜋
15.
sin−1 (sin ( 6 ))
16.
cos −1 (sin (− 6 ))
17.
sin−1 (tan ( 4 ))
18.
arcsin(0)
19.
cos −1 (cos ( 6 ))
20.
sin(cos−1 (−
𝜋
3𝜋
5𝜋
√3
))
2
√2
)
2
Additional Practice Part 3
√3
1.
sin−1 ( 2 )
2.
sin−1 (2)
3.
4.
tan−1 0
cos−1 1
5.
cos−1 ( 2 )
1
1
−1
6.
7.
tan 1
tan−1(−1)
8.
sin−1 (−
9.
√3
)
2
1
sin−1 (− )
√2
𝜋
10.
11.
12.
tan−1(− √3)
cos−1 0
sin−1(1)
19.
arcsin (cos ( 3 ))
20.
arccos (tan ( 4 ))
13.
cos(sin−1 ( 2 ))
1
14.
sin(tan−1( 1 ))
21.
22.
cos(tan−1 √3)
tan−1 (cos( 𝜋 ))
15.
sin−1 (cos ( ))
16.
17.
18.
𝜋
4
7𝜋
−1
cos (cos ( 4 ))
1
cos (2 sin−1 ( 2 ))
−1 (−1 ))
sin(tan
𝜋