inverse trigonometric functions
Module 7 : Investigation 9
MAT 170 | Precalculus
November 30, 2016
question 1
(a) Complete the ”function machine” diagrams below :
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
2
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
angle
measure
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
3
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
angle
measure
vert.
distance
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
4
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
angle
measure
horiz.
distance
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
5
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
angle
measure
horiz.
distance
slope
of terminal
ray
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
6
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
angle
measure
horiz.
distance
slope
of terminal
ray
(−∞, ∞)
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
7
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
angle
measure
horiz.
distance
slope
of terminal
ray
(−∞, ∞)
[−1, 1]
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
8
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
(−∞, ∞)
angle
measure
horiz.
distance
slope
of terminal
ray
(−∞, ∞)
[−1, 1]
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
9
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
(−∞, ∞)
[−1, 1]
angle
measure
horiz.
distance
slope
of terminal
ray
(−∞, ∞)
[−1, 1]
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
10
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
(−∞, ∞)
[−1, 1]
angle
measure
slope
of terminal
ray
horiz.
distance
(−∞, ∞)
θ ̸= π + kπ
2
[−1, 1]
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
11
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
(−∞, ∞)
[−1, 1]
angle
measure
slope
of terminal
ray
horiz.
distance
(−∞, ∞)
[−1, 1]
θ ̸= π + kπ
2
(−∞, ∞)
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
12
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
(−∞, ∞)
[−1, 1]
angle
measure
slope
of terminal
ray
horiz.
distance
(−∞, ∞)
[−1, 1]
θ ̸= π + kπ
2
(−∞, ∞)
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
None of these functions are one-to-one.
13
question 1
(a) Complete the ”function machine” diagrams below :
angle
measure
angle
measure
vert.
distance
(−∞, ∞)
[−1, 1]
angle
measure
slope
of terminal
ray
horiz.
distance
(−∞, ∞)
[−1, 1]
θ ̸= π + kπ
2
(−∞, ∞)
(d) We would like to define inverse sine, cosine, and tangent functions.
Considering the above, what problem do we have in doing so ? (Hint : What
property must a function possess in order to have an inverse ?)
None of these functions are one-to-one. For example,
1 = sin(π/2) = sin(5π/2) = sin(9π/2) = · · ·
14
inverse sine, cosine, and tangent functions
In order to make the sine, cosine, and tangent functions one-to-one, we will
restrict their domain.
15
inverse sine, cosine, and tangent functions
In order to make the sine, cosine, and tangent functions one-to-one, we will
restrict their domain.
1
−3π
−2π
−π
π
2π
3π
−1
sin (x) is restricted to [−π/2, π/2]
16
inverse sine, cosine, and tangent functions
In order to make the sine, cosine, and tangent functions one-to-one, we will
restrict their domain.
1
−3π
−2π
−π
π
2π
3π
−1
1
sin (x) is restricted to [−π/2, π/2]
cos (x) is restricted to [0, π]
−3π
−2π
−π
π
2π
3π
−1
17
inverse sine, cosine, and tangent functions
In order to make the sine, cosine, and tangent functions one-to-one, we will
restrict their domain.
1
−3π
−2π
−π
π
2π
−π
3π
− π2
π
2
π
−1
1
sin (x) is restricted to [−π/2, π/2]
cos (x) is restricted to [0, π]
−3π
−2π
−π
π
tan (x) is restricted to [−π/2, π/2]
2π
3π
−1
18
inverse sine, cosine, and tangent functions
In order to make the sine, cosine, and tangent functions one-to-one, we will
restrict their domain.
1
−3π
−2π
−π
π
2π
−π
3π
− π2
π
2
π
−1
1
sin (x) is restricted to [−π/2, π/2]
cos (x) is restricted to [0, π]
−3π
−2π
−π
π
tan (x) is restricted to [−π/2, π/2]
2π
3π
−1
These restrictions are used by
convention, but are by no means the
only possible choice.
19
inverse sine, cosine, and tangent functions
Definition
The inverse sine function is denoted arcsin(x) or sin−1 (x) and is defined by taking an input of a real number x in [−1, 1] and outputting
the unique angle θ in [−π/2, π/2] such that sin(θ) = x.
sin(θ) = x
⇐⇒
sin−1 (x) = θ.
Definition
The inverse cosine function is denoted arccos(x) or cos−1 (x) and is
defined by taking an input of a real number x in [−1, 1] and outputting
the unique angle θ in [0, π] such that cos(θ) = x.
cos(θ) = x
⇐⇒
cos−1 (x) = θ.
20
inverse sine, cosine, and tangent functions
Definition
The inverse tangent function takes an input of any real number x
and outputs the unique angle θ in the interval [−π/2, π/2] such that
tan(θ) = x. We denote the inverse tangent function by arctan(x) or
tan−1 (x).
tan(θ) = x ⇐⇒ tan−1 (x) = θ.
21
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
22
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
23
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
•
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
24
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
•
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
25
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
•
≈ 0.25 radians
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
26
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
y = 1.25
•
≈ 0.25 radians
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
27
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
•
y = 1.25
•
•
•
≈ 0.25 radians
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
28
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
•
y = 1.25
•
•
•
≈ 0.25 radians
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
29
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
y = 1.25
•
•
•
≈ 0.25 radians
≈ 0.25 radians
•
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25).
30
question 2
Demonstrate on the following circles how one could estimate the value of
sin−1 (0.25).
y = 0.25
•
•
y = 1.25
•
•
•
≈ 0.25 radians
≈ 0.25 radians
•
After you have demonstrated on the circle, use your calculator to evaluate
sin−1 (0.25). ≈ 0.25268 radians
31
question 3
Demonstrate on the following circles how one could estimate the value of
cos−1 (−0.75).
After you have demonstrated on the circle, use your calculator to evaluate
cos−1 (−0.75).
32
question 3
Demonstrate on the following circles how one could estimate the value of
cos−1 (−0.75).
After you have demonstrated on the circle, use your calculator to evaluate
cos−1 (−0.75). ≈ 2.418858 radians
33
question 4
Demonstrate on the following circles how one could estimate the value of
tan−1 (2).
After you have demonstrated on the circle, use your calculator to evaluate
tan−1 (2).
34
question 4
Demonstrate on the following circles how one could estimate the value of
tan−1 (2).
After you have demonstrated on the circle, use your calculator to evaluate
tan−1 (2). ≈ 1.10715 radians
35
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
36
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
37
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
Let θ1 be the angle in question between −π/2 and π/2.
38
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
Let θ1 be the angle in question between −π/2 and π/2.
Then we know 62 sin(θ1 ) = 42.
39
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
Let θ1 be the angle in question between −π/2 and π/2.
Then we know 62 sin(θ1 ) = 42. This means
sin(θ1 ) =
42
62
40
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
Let θ1 be the angle in question between −π/2 and π/2.
Then we know 62 sin(θ1 ) = 42. This means
( )
42
−1 42
sin(θ1 ) =
⇐⇒ θ1 = sin
≈ 0.74425 radians
62
62
41
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(a) What angle(s) has Savanna rotated from the 3 o’clock position
when she is 42 feet above the horizontal diameter of the Ferris
wheel ?
Let θ1 be the angle in question between −π/2 and π/2.
Then we know 62 sin(θ1 ) = 42. This means
( )
42
−1 42
sin(θ1 ) =
⇐⇒ θ1 = sin
≈ 0.74425 radians
62
62
The other angle θ2 is
θ2 = π − 0.74425 ≈ 2.397 radians
42
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
43
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
Let θ1 be the angle in question between 0 and π.
44
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
Let θ1 be the angle in question between 0 and π. Then we
know 62 cos(θ1 ) = −20.
45
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
Let θ1 be the angle in question between 0 and π. Then we
know 62 cos(θ1 ) = −20. This means
cos(θ1 ) = −
20
62
46
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
Let θ1 be the angle in question between 0 and π. Then we
know 62 cos(θ1 ) = −20. This means
(
)
20
20
−1
−
cos(θ1 ) = −
⇐⇒ θ1 = cos
≈ 1.89925 radians
62
62
47
question 5
Savanna is sitting in the bucket of a Ferris wheel at the 3 o’clock
position. The Ferris wheel has a radius of 62 feet and begins moving
counter-clockwise.
(b) What angle(s) has Savanna rotated from the 3 o’clock position
when she is -20 feet to the right of the vertical diameter of the
Ferris wheel ?
Let θ1 be the angle in question between 0 and π. Then we
know 62 cos(θ1 ) = −20. This means
(
)
20
20
−1
−
cos(θ1 ) = −
⇐⇒ θ1 = cos
≈ 1.89925 radians
62
62
The other angle θ2 is
θ2 = 2π − 1.89925 ≈ 4.384 radians
48
49