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Copyright © 2005 Pearson Education, Inc.
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Chapter 6 Inverse Circular Functions and Trigonometric Equations
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Copyright © 2005 Pearson Education, Inc. 6.1 Inverse Circular Functions
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Copyright © 2005 Pearson Education, Inc. Slide 6-4 Horizontal Line Test Any horizontal line will intersect the graph of a one-to-one function in at most one point. The inverse function of the one-to-one function f is defined as
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Copyright © 2005 Pearson Education, Inc. Slide 6-5 Inverse Functions Review 1. In a one-to-one function, each x-value corresponds to only one y-value and each y-value corresponds to only one x-value. 2. If a function f is one-to-one, then f has an inverse function f -1. 3. The domain of f is the range of f -1 and the range of f is the domain of f -1. 4. The graphs of f and f -1 are reflections of each other about the line y = x. 5. To find f -1 (x) from f(x), follow these steps. Replace f(x) with y and interchange x and y. Solve for y. Replace y with f -1 (x).
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Copyright © 2005 Pearson Education, Inc. Slide 6-6 Inverse Sine Function means that x = sin y, for Example: Find
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Copyright © 2005 Pearson Education, Inc. Slide 6-7 Examples sin 1 2 Not possible to evaluate because there is no angle whose sine is 2.
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Copyright © 2005 Pearson Education, Inc. Slide 6-8 Inverse Sine Function
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Copyright © 2005 Pearson Education, Inc. Slide 6-9 Inverse Cosine Function means that x = cos y, for Example: Find
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Copyright © 2005 Pearson Education, Inc. Slide 6-10 Inverse Cosine Function
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Copyright © 2005 Pearson Education, Inc. Slide 6-11 Inverse Tangent Function means that x = tan y, for
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Copyright © 2005 Pearson Education, Inc. Slide 6-12 Inverse Tangent Function
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Copyright © 2005 Pearson Education, Inc. Slide 6-13 Other Inverse Functions
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Copyright © 2005 Pearson Education, Inc. Slide 6-14 Examples Find the degree measure of in the following. a) = arctan 1b) = sec 1 2 a) must be in ( 90 , 90 ), since 1 > 0, must be in quadrant I. The alternative statement, tan = 1, leads to = 45 . b) sec = 2, for sec 1 x, is in quadrant I or II. Because 2 is positive, is in quadrant I and sec 60 = 2.
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Copyright © 2005 Pearson Education, Inc. Slide 6-15 Example Find y in radians if y = arctan( 6.24). Calculator in radian mode Enter tan 1 ( 6.24) y 1.411891065 Find y in radians if y = arccos 2. Calculator in radian mode Enter cos 1 (2) (error message since the domain of the inverse cosine function is [ 1, 1].
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Copyright © 2005 Pearson Education, Inc. Slide 6-16 Example: Evaluate the expression without using a calculator. Let The inverse tangent function yields values only in quadrants I and III, since 3/2 is positive, is in quadrant I.
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Copyright © 2005 Pearson Education, Inc. Slide 6-17 Example: Evaluate the expression without using a calculator continued Sketch and label the triangle. The hypotenuse is
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Copyright © 2005 Pearson Education, Inc. Slide 6-18 Example Evaluate the expression without using a calculator. Let arcsin 2/5 = B Since arcsin 2/5 = B, sin B = 2/5. Sketch a triangle in quadrant I, find the length of the third side, then find tan B.
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Copyright © 2005 Pearson Education, Inc. Slide 6-19 Example continued
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Copyright © 2005 Pearson Education, Inc. Slide 6-20 Homework Do pp. 246-249, 1-6 all, 8, 14-52 even, 58-62 all, 64-78 even, 79-82 all, & 84-96 even
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