Lect21 - Inverse Trigonometric Functions: We'd like to...

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92.131 Lecture 23 1 of 20 Inverse Trigonometric Functions: We’d like to explore the question of inverses of the sine, tangent, and secant functions. We’ll start with x x f sin ) ( = . Recall the graph:
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92.131 Lecture 23 2 of 20 Well, we can consider the “restricted sine function,” defined as x x f sin ) ( = for 2 , 2 π x , whose graph is This function does have an inverse; it is written as x arcsin or x 1 sin , and its graph is obtained by rotating the graph about the line x y = . ( ) 1 , 2 () 1 , 2
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92.131 Lecture 23 3 of 20 ( ) 1 , 2 π () 1 , 2 ( ) 2 , 1 ( ) 2 , 1 x x y 1 sin arcsin = =
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92.131 Lecture 23 4 of 20 Remarks: 1. We reserve the notation x 1 sin to mean the inverse sine function, NOT x x csc sin 1 = . 2. The domain of x 1 sin is ] 1 , 1 [ , and its range is 2 , 2 π , as can be seen from the graph. 3. Recall that 3 x y = is the inverse of 3 ) ( x x f = , and was defined as the number whose cube is x , that is, the number y such that x y y f = = 3 ) (. I n the same way, x y 1 sin = is the number y such that x y y f = = sin ) ( ; but notice that y is restricted to 2 , 2 , and so x 1 sin is the number in 2 , 2 whose sine is x . In most cases, it is more convenient to think of x 1 sin as the angle (in radians) in the 1 st or 4 th quadrant whose sine is x .
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92.131 Lecture 23 5 of 20 4. We chose to restrict the sine function to 2 , 2 π . We could have chosen 2 3 , 2 , but not ] , 0 [ . Example: Find if possible: a) 0 sin 1 b ) 1 sin 1 c) 2 1 arcsin d ) 2 sin 1 a) 0 0 sin 1 = from the graph. b) 2 1 sin 1 = from the graph.
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92.131 Lecture 23 6 of 20 c) 2 1 arcsin = the angle in the 1 st or 4 th quadrant whose sine is 2 1 . Consider the unit circle shown below. We recognize the 30 ° /60 ° /90 ° triangle: and so 6 2 1 arcsin π = .
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Lect21 - Inverse Trigonometric Functions: We'd like to...

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