4.7 Inverse Trigonometric Functions
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1 4.7 Inverse Trigonometric Functions Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 1 / 19
2 4.7 Inverse Trigonometric Functions 1 Inverse Sine Function Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 2 / 19
3 4.7 Inverse Trigonometric Functions 1 Inverse Sine Function 2 Other Inverse Trigonometric Functions Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 2 / 19
4 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
5 Today s Learning Target(s) 1 I can evaluate the inverse sine, inverse cosine, and inverse tangent functions. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 3 / 19
6 The Inverse Sine Function The inverse sine function is typically denoted by either arcsin or sin 1. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 4 / 19
7 The Inverse Sine Function The inverse sine function is typically denoted by either arcsin or sin 1. If you are to evaluate arcsin 2 2 or sin (these are the same thing), then you are trying to determine what value for θ satisfies the equation sin θ = 2 2. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 4 / 19
8 The Inverse Sine Function The inverse sine function is typically denoted by either arcsin or sin 1. If you are to evaluate arcsin 2 2 or sin (these are the same thing), then you are trying to determine what value for θ satisfies the equation sin θ = 2 2. In other words, you are trying to determine what angle gives you the sine described by the function. In this case, since sin θ = 2 2 when θ = π 4, then it follows that arcsin 2 2 = sin = π 4. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 4 / 19
9 Domain of the Inverse Sine Function Quick Question What is the domain of y = arcsin x = sin 1 x? In other words, what x-values can you put into the function? Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 5 / 19
10 Domain of the Inverse Sine Function Quick Question What is the domain of y = arcsin x = sin 1 x? In other words, what x-values can you put into the function? Quick Question What is the range of y = arcsin x = sin 1 x? In other words, what y-values can you get out of the function? Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 5 / 19
11 Evaluating the Inverse Sine Function Practice If possible, find each exact value. a arcsin ( ) 1 2 b sin 1 3 c sin 1 ( 1) Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 6 / 19
12 Inverse Cosine and Inverse Tangent Functions Both cosine and tangent have inverse functions defined very similarly to arcsin and sin 1. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 7 / 19
13 Inverse Cosine and Inverse Tangent Functions Both cosine and tangent have inverse functions defined very similarly to arcsin and sin 1. The inverse cosine function can be denoted by arccos or cos 1. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 7 / 19
14 Inverse Cosine and Inverse Tangent Functions Both cosine and tangent have inverse functions defined very similarly to 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
15 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
16 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
17 Evaluating the Inverse Trigonometric Functions Practice Find the exact value. a arccos 3 2 b cos 1 ( 0.5) c arctan(1) ( 3 ) d tan 1 3 Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 10 / 19
18 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
19 4.7 Inverse Trigonometric Functions (Part 1 of 2) Assignment Part 1: pg. 349 Vocabulary Check #1-3 Exercises #2-16 even, even Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 12 / 19
20 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
21 Quick Snow Day Review Practice What are the domain and range for each of y = arcsin x, y = arccos x, and y = arctan x? Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 14 / 19
22 Inverse Properties of Functions Inverse Properties of Functions If 1 x 1 and π 2 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 π 2 y π 2, then tan(arctan x) = x and arctan(tan y) = y. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 15 / 19
23 Inverse Properties of Functions Inverse Properties of Functions If 1 x 1 and π 2 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 π 2 y π 2, then tan(arctan x) = x and arctan(tan y) = y. The biggest takeaway from this is that the two functions cancel each other out, only if you are working within the domain of the innermost function. Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 15 / 19
24 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
25 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
26 Evaluating Compositions of Functions Practice Find the exact value. a cos [ arctan ( 3 )] 4 b sin [ arccos ( )] 2 3 Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 18 / 19
27 4.7 Inverse Trigonometric Functions (Part 2 of 2) Assignment Part 1: pg. 349 Vocabulary Check #1-3; Exercises #2-16 even, even Part 2: pg Exercises #37-40, even, 91-92, 94, Inverse Trigonometric Functions Assignment pg Vocabulary Check #1-3; Exercises #2-16 even, even, 37-40, even, 91-92, 94, 96 Accelerated Pre-Calculus 4.7 Inverse Trigonometric Functions Mr. Niedert 19 / 19
D. Correct! 2 does not belong to the domain of arcsin x.
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