Slide 4- 7 Homework, Page 421
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Homework, Page 421
Find the exact value.
1.
sin
1
sin
1
3
2
3
or 60
2 3
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Slide 4- 1
Homework, Page 421
Find the exact value.
5.
1
cos
2
1 1
cos
or 60
2 3
1
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Slide 4- 2
Homework, Page 421
Find the exact value.
1
9. sin
2
1
7
1
sin
4 4
2
45 315
1
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Slide 4- 3
Homework, Page 421
Use a calculator to find the approximate value. Express the
answer in degrees.
1
sin
0.362
13.
sin 1 0.362 21.223
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Slide 4- 4
Homework, Page 421
Use a calculator to find the approximate value. Express the
answer in radians.
1
tan
2.37
17.
tan 1 2.37 1.172 rad
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Slide 4- 5
Homework, Page 421
Describe the end behavior of the function.
21. y tan 1 x 2
lim tan
1
x
lim tan
x
x 2
1
2
x 2
2
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Slide 4- 6
Homework, Page 421
Find the exact value without a calculator.
sin cos
4
1
25.
2
1
sin cos sin
4
2 4
1
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Slide 4- 7
Homework, Page 421
Find the exact value without a calculator.
29. arcsin cos
3
1
arcsin cos arcsin
3
2 6
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Slide 4- 8
Homework, Page 421
Analyze each function for domain, range, continuity, increasing
or decreasing, symmetry, boundedness, extrema, asymptotes, and
end behavior.
33.
f x sin 1 x
Domain:x : 1 x 1; Range:y :
y
;
2
2
Continuous on domain; Increasing on domain; Symmetrical
about the origin; Bounded; Absolute minimum at x = 1;
Absolute maximum at x = 1; No asymptotes;
Since bounded, no end behavior
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Slide 4- 9
Homework, Page 421
Use transformations to describe how the graph of the function is
related to a basic inverse trigonometric graph. State the domain
and range.
37. f x sin 1 2 x
To form the graph of f x sin
1
2 x from the graph of
1
f x sin x , apply a horizontal stretch of .
2
1
1
Domain : x : x
2
2
1
Range : y :
2
y
2
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Slide 4- 10
Homework, Page 421
Find an exact solution to the equation without a calculator.
1
41. sin sin x 1
sin sin 1 x 1 x 1
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Slide 4- 11
Homework, Page 421
Find an exact solution to the equation without a calculator.
1
45. cos cos x 3
1
1
1
cos cos x x
3
3
1
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Slide 4- 12
Homework, Page 421
Find an algebraic expression equivalent to the given expression.
49.
tan arcsin x
tan arcsin x
x
1 x2
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Slide 4- 13
Homework, Page 421
53.
The bottom of a picture is 2 ft above eye level and the
picture is 12 ft tall. Angle θ is formed by the lines of vision to
the top and bottom of the picture.
14
1 2
tan tan
x
x
1 14
1 2
1 tan ; 2 tan
x
x
1 14
1 2
1 2 tan tan
x
x
a. Show that
1
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Slide 4- 14
Homework, Page 421
53.
b.
Graph θ in a [0, 25] by [0, 50] viewing window
using degree mode. Show that the maximum value of θ occurs
approximately 5.3 ft from the picture.
c.
How far, to the nearest foot, are you standing if θ =35º?
2 ft or 15 ft
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Slide 4- 15
57.
Homework, Page 421
sin sin 1 x x for all real numbers x. Justify your
answer.
False, the function sin 1 x has a restricted domain of
1 x 1.
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Slide 4- 16
Homework, Page 421
1
sec
tan
x
61.
a.
b.
x
csc x
c 1 x 1 x
1
cos
sec 1 x 2
1 x2
2
c.
1 x
x
sec tan x tan x
1
1
2
d.
1 x2
sin x
e.
cos x
2
2
2
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Slide 4- 17
Homework, Page 421
65. Graph each of the three functions and interpret the graph to
find domain, range, and period. Which of these functions has
points of discontinuity? Are the discontinuities removable or
nonremovable?
a. y sin 1 sin x
The function has a domain
of all real numbers, a range
of –π/2 to π/2, and a period
of 2π.
4 ,4 by ,
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Slide 4- 18
Homework, Page 421
65. b. y cos 1 cos x
The function has domain of all
real numbers, a range of 0 to π,
and a period of 2π.
c. y tan tan x
The function has domain of all
real numbers ≠ nπ/2, n≠0, a
range of –π/2 to π/2, and a
period of π. The
discontinuities are not
removable.
1
4 ,4 by ,
4 ,4 by ,
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Homework, Page 421
69. Use elementary transformations and the arctangent
function to construct a function with a domain of all real
numbers that has horizontal asymptotes at y = 24 and
y = 42.
y tan 1 x y tan 1 x 33
18
A a A9
2
18 1
y tan x 33
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Slide 4- 20
4.8
Solving Problems with Trigonometry
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Quick Review
1. Solve for a.
a
23º
3
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Slide 4- 22
Quick Review
2. Find the complement of 47 .
3. Find the supplement of 47 .
4. State the bearing that describes the direction NW (northwest).
5. State the amplitude and period of the sinusoid 3cos 2( x 1).
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Slide 4- 23
Quick Review Solutions
1. Solve for a.
a
23º
7.678
3
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Slide 4- 24
Quick Review Solutions
2. Find the complement of 47 . 43
3. Find the supplement of 47 . 133
4. State the bearing that describes the direction NW (northwest).
5. State the amplitude and period of the sinusoid 3cos 2( x 1).
A 3, p
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135
Slide 4- 25
What you’ll learn about
More Right Triangle Problems
Simple Harmonic Motion
… and why
These problems illustrate some of the betterknown applications of trigonometry.
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Slide 4- 26
Angle of Elevation, Angle of Depression
An angle of elevation is the angle through which the eye moves
up from horizontal to look at something above. An angle of
depression is the angle through which the eye moves down from
horizontal to look at something below.
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Slide 4- 27
Example Using Angle of Elevation
The angle of elevation from the buoy to the top of the Barnegat
Bay lighthouse 130 feet above the surface of the water is 5º. Find
the distance x from the base of the lighthouse to the buoy.
130
5º
x
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Example Using Angle of Depression
An observer watches a car approach from the top of a 100-ft
building. If the angle of depression of the car changes from 15º to
35º during the period of observation, how far did the car move?
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Example Navigation Problem
A hydrofoil travels at a speed of 40 knots from Fort Lauderdale on
a course of 065º for two hours and then turns to a course of 155º for
four more hours. What are the direction and distance from Fort
Lauderdale to the hydrofoil after the six hours?
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Example Navigation Problem
Port A is directly east of Port B on Lake Wonderful. A police boat
leaves Port B at 23 kts on a course of 095ºand a smuggler’s boat
leaves Port A on a course of 195º at the same time. Two hours
later the boats collide. How fast was the smuggler’s boat traveling?
How far is port A from Port B?
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Simple Harmonic Motion
A point moving on a number line is in simple harmonic
motion if its directed distance d from the origin is given
by either d a sin t or d a cos t , where a and
are real numbers and 0. The motion has frequency
/ 2 , which is the number of oscillations per unit of time.
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Example Calculating Harmonic Motion
A mass oscillating up and down on the bottom of a spring
(assuming perfect elasticity and no friction or air
resistance) can be modeled as harmonic motion. If the
weight is displaced a maximum of 4 cm, find the modeling
equation if it takes 3 seconds to complete one cycle.
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Homework
Homework Assignment #8
Review Section 4.8
Page 431, Exercises: 1 – 49 (EOO)
Quiz next time
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