Algebra and Trigonometry 2e2-11

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Real-World Applications
ⓐ Graph on the interval
ⓑ Find and interpret the stretching factor,
period, and asymptote.
ⓒ Evaluate and and discuss the
function’s values at those inputs.
54. The function marks the
distance in the movement of a light beam from a
police car across a wall for time in seconds, and
distance in feet.
ⓐ What is a reasonable domain for
ⓑ Graph on this domain.
ⓒ Find and discuss the meaning of any vertical
asymptotes on the graph of
ⓓ Calculate and interpret Round to the
second decimal place.
ⓔ Calculate and interpret Round to the second
decimal place.
ⓕ What is the minimum distance between the
fisherman and the boat? When does this occur?
55. Standing on the shore of a lake, a fisherman sights a
boat far in the distance to his left. Let measured in
radians, be the angle formed by the line of sight to the
ship and a line due north from his position. Assume due
north is 0 and is measured negative to the left and
positive to the right. (See Figure 19.) The boat travels
from due west to due east and, ignoring the curvature
of the Earth, the distance in kilometers, from the
fisherman to the boat is given by the function
Figure 19
ⓐ Graph on the interval
ⓑ Evaluate and interpret the information.
ⓒ What is the minimum distance between the
comet and Earth? When does this occur? To which
constant in the equation does this correspond?
ⓓ Find and discuss the meaning of any vertical
asymptotes.
56. A laser rangefinder is locked on a comet
approaching Earth. The distance in
kilometers, of the comet after days, for in the
interval 0 to 30 days, is given by
8.2 • Graphs of the Other Trigonometric Functions 799
ⓐ Write a function expressing the altitude
in miles, of the rocket above the ground after
seconds. Ignore the curvature of the Earth.
ⓑ Graph on the interval
ⓒ Evaluate and interpret the values and
ⓓ What happens to the values of as
approaches 60 seconds? Interpret the meaning of
this in terms of the problem.
57. A video camera is focused on a rocket on a
launching pad 2 miles from the camera. The angle
of elevation from the ground to the rocket after
seconds is
8.3 Inverse Trigonometric Functions
Learning Objectives
In this section, you will:
Understand and use the inverse sine, cosine, and tangent functions.
Find the exact value of expressions involving the inverse sine, cosine, and tangent functions.
Use a calculator to evaluate inverse trigonometric functions.
Find exact values of composite functions with inverse trigonometric functions.
For any right triangle, given one other angle and the length of one side, we can figure out what the other angles and
sides are. But what if we are given only two sides of a right triangle? We need a procedure that leads us from a ratio of
sides to an angle. This is where the notion of an inverse to a trigonometric function comes into play. In this section, we
will explore the inverse trigonometric functions.
Understanding and Using the Inverse Sine, Cosine, and Tangent Functions
In order to use inverse trigonometric functions, we need to understand that an inverse trigonometric function “undoes”
what the original trigonometric function “does,” as is the case with any other function and its inverse. In other words,
the domain of the inverse function is the range of the original function, and vice versa, as summarized in Figure 1.
Figure 1
For example, if then we would write Be aware that does not mean The
following examples illustrate the inverse trigonometric functions:
• Since then
• Since then
• Since then
In previous sections, we evaluated the trigonometric functions at various angles, but at times we need to know what
angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that, for a one-to-
one function, if then an inverse function would satisfy
Bear in mind that the sine, cosine, and tangent functions are not one-to-one functions. The graph of each function would
fail the horizontal line test. In fact, no periodic function can be one-to-one because each output in its range corresponds
to at least one input in every period, and there are an infinite number of periods. As with other functions that are not
one-to-one, we will need to restrict the domain of each function to yield a new function that is one-to-one. We choose a
domain for each function that includes the number 0. Figure 2 shows the graph of the sine function limited to
and the graph of the cosine function limited to
800 8 • Periodic Functions
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Figure 2 (a) Sine function on a restricted domain of (b) Cosine function on a restricted domain of
Figure 3 shows the graph of the tangent function limited to
Figure 3 Tangent function on a restricted domain of
These conventional choices for the restricted domain are somewhat arbitrary, but they have important, helpful
characteristics. Each domain includes the origin and some positive values, and most importantly, each results in a one-
to-one function that is invertible. The conventional choice for the restricted domain of the tangent function also has the
useful property that it extends from one vertical asymptote to the next instead of being divided into two parts by an
asymptote.
On these restricted domains, we can define the inverse trigonometric functions.
• The inverse sine function means The inverse sine function is sometimes called the arcsine
function, and notated
• The inverse cosine function means The inverse cosine function is sometimes called the
arccosine function, and notated
• The inverse tangent function means The inverse tangent function is sometimes called the
arctangent function, and notated
∞
The graphs of the inverse functions are shown in Figure 4, Figure 5, and Figure 6. Notice that the output of each of these
inverse functions is a number, an angle in radian measure. We see that has domain and range
has domain and range and has domain of all real numbers and range To find the
domain and range of inverse trigonometric functions, switch the domain and range of the original functions. Each graph
of the inverse trigonometric function is a reflection of the graph of the original function about the line
8.3 • Inverse Trigonometric Functions 801
Figure 4 The sine function and inverse sine (or arcsine) function
Figure 5 The cosine function and inverse cosine (or arccosine) function
Figure 6 The tangent function and inverse tangent (or arctangent) function
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...
Relations for Inverse Sine, Cosine, and Tangent Functions
For angles in the interval if then
For angles in the interval if then
For angles in the interval if then
EXAMPLE 1
Writing a Relation for an Inverse Function
Given write a relation involving the inverse sine.
Solution
Use the relation for the inverse sine. If then .
In this problem, and
TRY IT #1 Given write a relation involving the inverse cosine.
Finding the Exact Value of Expressions Involving the Inverse Sine, Cosine, and
Tangent Functions
Now that we can identify inverse functions, we will learn to evaluate them. For most values in their domains, we must
evaluate the inverse trigonometric functions by using a calculator, interpolating from a table, or using some other
numerical technique. Just as we did with the original trigonometric functions, we can give exact values for the inverse
functions when we are using the special angles, specifically (30°), (45°), and (60°), and their reflections into other
quadrants.
HOW TO
Given a “special” input value, evaluate an inverse trigonometric function.
1. Find angle for which the original trigonometric function has an output equal to the given input for the inverse
trigonometric function.
2. If is not in the defined range of the inverse, find another angle that is in the defined range and has the same
sine, cosine, or tangent as depending on which corresponds to the given inverse function.
EXAMPLE 2
Evaluating Inverse Trigonometric Functions for SpecialInput Values
Evaluate each of the following.
ⓐ ⓑ ⓒ ⓓ
Solution
ⓐ Evaluating is the same as determining the angle that would have a sine value of In other words,
what angle would satisfy There are multiple values that would satisfy this relationship, such as and
but we know we need the angle in the interval so the answer will be Remember that
the inverse is a function, so for each input, we will get exactly one output.
8.3 • Inverse Trigonometric Functions 803