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4.7 – Inverse Trigonometric Functions
Accelerated Pre-Calculus
Mr. Niedert
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 1 / 19
4.7 – Inverse Trigonometric Functions
1 Inverse Sine Function
2 Other Inverse Trigonometric Functions
3 Composition of Functions
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 2 / 19
4.7 – Inverse Trigonometric Functions
1 Inverse Sine Function
2 Other Inverse Trigonometric Functions
3 Composition of Functions
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 2 / 19
4.7 – Inverse Trigonometric Functions
1 Inverse Sine Function
2 Other Inverse Trigonometric Functions
3 Composition of Functions
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 2 / 19
Today’s Learning Target(s)
1 I can evaluate the inverse sine, inverse cosine, and inverse tangentfunctions.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 3 / 19
The Inverse Sine Function
The inverse sine function is typically denoted by either arcsin or sin−1.
If you are to evaluate arcsin√22 or sin−1
√22 (these are the same
thing), then you are trying to determine what value for θ satisfies the
equation sin θ =√22 .
In other words, you are trying to determine what angle gives you the
sine described by the function. In this case, since sin θ =√22 when
θ = π4 , then it follows that arcsin
√22 = sin−1
√22 = π
4 .
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 4 / 19
The Inverse Sine Function
The inverse sine function is typically denoted by either arcsin or sin−1.
If you are to evaluate arcsin√22 or sin−1
√22 (these are the same
thing), then you are trying to determine what value for θ satisfies the
equation sin θ =√22 .
In other words, you are trying to determine what angle gives you the
sine described by the function. In this case, since sin θ =√22 when
θ = π4 , then it follows that arcsin
√22 = sin−1
√22 = π
4 .
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 4 / 19
The Inverse Sine Function
The inverse sine function is typically denoted by either arcsin or sin−1.
If you are to evaluate arcsin√22 or sin−1
√22 (these are the same
thing), then you are trying to determine what value for θ satisfies the
equation sin θ =√22 .
In other words, you are trying to determine what angle gives you the
sine described by the function. In this case, since sin θ =√22 when
θ = π4 , then it follows that arcsin
√22 = sin−1
√22 = π
4 .
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 4 / 19
Domain of the Inverse Sine Function
Quick Question
What is the domain of y = arcsin x = sin−1 x? In other words, whatx-values can you “put into” the function?
Quick Question
What is the range of y = arcsin x = sin−1 x? In other words, whaty -values can you “get out of” the function?
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 5 / 19
Domain of the Inverse Sine Function
Quick Question
What is the domain of y = arcsin x = sin−1 x? In other words, whatx-values can you “put into” the function?
Quick Question
What is the range of y = arcsin x = sin−1 x? In other words, whaty -values can you “get out of” the function?
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 5 / 19
Evaluating the Inverse Sine Function
Practice
If possible, find each exact value.
a arcsin(12
)b sin−1
√3
c sin−1(−1)
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 6 / 19
Inverse Cosine and Inverse Tangent Functions
Both cosine and tangent have inverse functions defined very similarlyto arcsin and sin−1.
The inverse cosine function can be denoted by arccos or cos−1.
The inverse tangent function can be denoted by arctan or tan−1.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 7 / 19
Inverse Cosine and Inverse Tangent Functions
Both cosine and tangent have inverse functions defined very similarlyto arcsin and sin−1.
The inverse cosine function can be denoted by arccos or cos−1.
The inverse tangent function can be denoted by arctan or tan−1.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 7 / 19
Inverse Cosine and Inverse Tangent Functions
Both cosine and tangent have inverse functions defined very similarlyto arcsin and sin−1.
The inverse cosine function can be denoted by arccos or cos−1.
The inverse tangent function can be denoted by arctan or tan−1.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 7 / 19
Domain and Range of the Cosine Function
Quick Question
What would the domain and range have to be for y = arccos x = cos−1 x?
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 8 / 19
Domain and Range of the Tangent Function
Quick Question
What would the domain and range have to be for y = arctan x = tan−1 x?
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 9 / 19
Evaluating the Inverse Trigonometric Functions
Practice
Find the exact value.
a arccos√32
b cos−1(−0.5)
c arctan(1)
d tan−1(√
33
)
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 10 / 19
Calculators and Inverse Trigonometric Functions
Example
Use a calculator to approximate the value (if possible).
a arctan 4.84
b arccos(−0.349)
c sin−1(−1.1)
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 11 / 19
4.7 – Inverse Trigonometric Functions (Part 1 of 2)Assignment
Part 1: pg. 349 Vocabulary Check #1-3 Exercises #2-16 even, 20-34 even
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 12 / 19
Today’s Learning Target(s)
I can use the inverse properties to evaluate composition functions.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 13 / 19
Quick Snow Day Review
Practice
What are the domain and range for each of y = arcsin x , y = arccos x , andy = arctan x?
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 14 / 19
Inverse Properties of Functions
Inverse Properties of Functions
If −1 ≤ x ≤ 1 and −π2 ≤ y ≤ π
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 .
If x ∈ R and −π2 ≤ y ≤ π
2 , then
tan(arctan x) = x and arctan(tan y) = y .
The biggest takeaway from this is that the two functions cancel eachother out, only if you are working within the domain of the innermostfunction.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 15 / 19
Inverse Properties of Functions
Inverse Properties of Functions
If −1 ≤ x ≤ 1 and −π2 ≤ y ≤ π
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 .
If x ∈ R and −π2 ≤ y ≤ π
2 , then
tan(arctan x) = x and arctan(tan y) = y .
The biggest takeaway from this is that the two functions cancel eachother out, only if you are working within the domain of the innermostfunction.
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 15 / 19
Using Inverse Properties
Practice
If possible, find the exact value.
a tan [arctan(−5)]
b arcsin(sin 5π
3
)c cos
(cos−1 π
)
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 16 / 19
Evaluating Compositions of Functions
Example
Find the exact value.
a tan(arccos 2
3
)b cos
[arcsin
(−3
5
)]
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 17 / 19
Evaluating Compositions of Functions
Practice
Find the exact value.
a cos[arctan
(−3
4
)]b sin
[arccos
(23
)]
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 18 / 19
4.7 – Inverse Trigonometric Functions (Part 2 of 2)Assignment
Part 1: pg. 349 Vocabulary Check #1-3; Exercises #2-16 even, 20-34evenPart 2: pg. 349-351 Exercises #37-40, 44-58 even, 91-92, 94, 96
4.7 – Inverse Trigonometric Functions Assignmentpg. 349-351 Vocabulary Check #1-3; Exercises #2-16 even, 20-34 even,37-40, 44-58 even, 91-92, 94, 96
Accelerated Pre-Calculus 4.7 – Inverse Trigonometric Functions Mr. Niedert 19 / 19
