Copyright © 2005 Pearson Education, Inc.. Chapter 6 Inverse Circular Functions and Trigonometric...
20
Copyright © 2005 Pearson Education, Inc.
-
Upload
ella-underwood -
Category
Documents
-
view
218 -
download
0
Transcript of Copyright © 2005 Pearson Education, Inc.. Chapter 6 Inverse Circular Functions and Trigonometric...
Copyright © 2005 Pearson Education, Inc.
Copyright © 2005 Pearson Education, Inc.
Chapter 6
Inverse Circular Functions and Trigonometric Equations
Copyright © 2005 Pearson Education, Inc.
6.1
Inverse Circular Functions
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
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-1and 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).
Copyright © 2005 Pearson Education, Inc. Slide 6-6
Inverse Sine Function
means that x = sin y, for
Example: Find
Copyright © 2005 Pearson Education, Inc. Slide 6-7
Examples
sin1 2
Not possible to evaluate because there is no angle whose sine is 2.
Copyright © 2005 Pearson Education, Inc. Slide 6-8
Inverse Sine Function
Copyright © 2005 Pearson Education, Inc. Slide 6-9
Inverse Cosine Function
means that x = cos y, for
Example: Find
Copyright © 2005 Pearson Education, Inc. Slide 6-10
Inverse Cosine Function
Copyright © 2005 Pearson Education, Inc. Slide 6-11
Inverse Tangent Function
means that x = tan y, for
Copyright © 2005 Pearson Education, Inc. Slide 6-12
Inverse Tangent Function
Copyright © 2005 Pearson Education, Inc. Slide 6-13
Other Inverse Functions
Copyright © 2005 Pearson Education, Inc. Slide 6-14
Examples
Find the degree measure of in the following. a) = arctan 1 b) = sec1 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 sec1 x, is in quadrant I or II. Because 2 is positive, is in quadrant I and sec 60 = 2.
Copyright © 2005 Pearson Education, Inc. Slide 6-15
Example
Find y in radians if y = arctan(6.24). Calculator in radian mode Enter tan1(6.24) y 1.411891065
Find y in radians if y = arccos 2. Calculator in radian mode Enter cos1(2) (error message since the domain of
the inverse cosine function is [1, 1].
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.
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
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.
Copyright © 2005 Pearson Education, Inc. Slide 6-19
Example continued
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
