20
Chapter 1 Right Triangle Trigonometry
1.3
Exercises
1. From a position 150 ft above the ground, an observer in a building measures angles of depression of 12 and 34 to the top and
bottom, respectively, of a smaller building, as in the picture on
the ri
Inverse Trigonometric Functions Section 5.3
125
Thus, y = tan1 x is a function whose domain is the set of all real numbers and whose
range is the interval , . In other words:
22
tan1 (tan y) = y
for < y <
2
tan (tan1 x) = x
2
(5.6)
for all real x
(5.7)
Ex
Trigonometric Functions of Any Angle Section 1.4
25
We can now dene the trigonometric functions of any angle in terms of Cartesian coordinates. Recall that the x y-coordinate plane consists of points denoted by pairs ( x, y) of
real numbers. The rst numbe
124
Chapter 5 Graphing and Inverse Functions
5.3
The inverse tangent function y = tan1 x (sometimes called the arc tangent and denoted by y = arctan x) can be determined similarly. The function y = tan x is one-to-one over
the interval , , as we see in Fi
24
Chapter 1 Right Triangle Trigonometry
1.4
1.4 Trigonometric Functions of Any Angle
To dene the trigonometric functions of any angle - including angles less than 0 or greater
than 360 - we need a more general denition of an angle. We say that an angle i
Applications and Solving Right Triangles Section 1.3
25. A manufacturer needs to place ten identical ball bearings against
the inner side of a circular container such that each ball bearing touches two other ball bearings, as in the picture on the
right.
Inverse Trigonometric Functions Section 5.3
123
cos1 (cos y) = y
for 0 y
(5.4)
cos (cos
for 1 x 1
(5.5)
1
x) = x
The graph of y = cos1 x is shown below in Figure 5.3.7. Notice the symmetry about the
line y = x with the graph of y = cos x.
y
y = cos1 x
1
22
Chapter 1 Right Triangle Trigonometry
1.3
24. In Example 1.10 in Section 1.2, we found the exact values of all six trigonometric functions of
75 . For example, we showed that cot 75 = 6 2 . So since tan 15 = cot 75 by the Cofunction
6+ 2
Theorem, this
122
Chapter 5 Graphing and Inverse Functions
5.3
Example 5.14 illustrates an important point: sin1 x should always be a number between
and . If you get a number outside that range, then you made a mistake somewhere.
2
2
sin
This why in Example 1.27 in Se
Applications and Solving Right Triangles Section 1.3
8. A ball bearing sits between two metal grooves, with the top groove having an
angle of 120 and the bottom groove having an angle of 90 , as in the picture
on the right. What must the diameter of the b
Inverse Trigonometric Functions Section 5.3
121
The basic idea is that f 1 undoes what f does, and vice versa. In other words,
f 1 ( f ( x) = x
for all x in the domain of f , and
f (f
for all y in the range of f .
1
( y) = y
We know from their graphs that
120
Chapter 5 Graphing and Inverse Functions
5.3
5.3 Inverse Trigonometric Functions
We have briey mentioned the inverse
trigonometric functions before, for example in Section
1.3 when we discussed how to use the 1 1 and 1 buttons on a calculator to nd
s
Applications and Solving Right Triangles Section 1.3
19
For some problems it may help to remember that when a right trir
angle has a hypotenuse of length r and an acute angle , as in the
r sin
picture on the right, the adjacent side will have length r co
Properties of Graphs of Trigonometric Functions Section 5.2
19. y = 3 sin x 5 cos x
20. y = 5 sin 3 x + 12 cos 3 x
119
21. y = 2 cos x + 2 sin x
22. Find the amplitude of the function y = 2 sin ( x2 ) + cos ( x2 ).
For Exercises 23-25, nd the period of th
18
Chapter 1 Right Triangle Trigonometry
1.3
Example 1.18
A slider-crank mechanism is shown in Figure 1.3.2 below. As the piston moves downward the connecting rod rotates the crank in the clockwise direction, as indicated.
piston
a
A
e
conn
b
C
c
ro
cting
118
Chapter 5 Graphing and Inverse Functions
5.2
In engineering two periodic functions with the same period are said to be out of phase if
their phase shifts differ. For example, sin x and sin x would be radians (or 30 ) out of
6
6
phase, and sin x would
Applications and Solving Right Triangles Section 1.3
17
You may have noticed that the solutions to the examples we have shown required at least
one right triangle. In applied problems it is not always obvious which right triangle to
use, which is why thes
Properties of Graphs of Trigonometric Functions Section 5.2
117
The phase shift is dened similarly for the other trigonometric functions.
Example 5.11
Find the amplitude, period, and phase shift of y = 3 cos (2 x ).
2
2
Solution: The amplitude is 3, the p
16
Chapter 1 Right Triangle Trigonometry
1.3
Example 1.15
As another application of trigonometry to astronomy, we will nd the distance
from the earth to the sun. Let O be the center of the earth, let A be a point
on the equator, and let B represent an obj
116
Chapter 5 Graphing and Inverse Functions
5.2
Generalizing Example 5.9, an expression of the form a sin x + b cos x is equivalent
a
to a2 + b2 sin ( x + ), where is an angle such that cos = 2 2 and sin = 2b 2 . So
a +b
a +b
y = a sin x + b cos x will h
Applications and Solving Right Triangles Section 1.3
15
Example 1.13
A blimp 4280 ft above the ground measures an angle of depression of 24 from its horizontal line of
sight to the base of a house on the ground. Assuming the ground is at, how far away alo
Properties of Graphs of Trigonometric Functions Section 5.2
115
In general, a combination of sines and cosines will have a period equal to the lowest common multiple of the periods of the sines and cosines being added. In Example 5.9, sin x
and cos x each
14
Chapter 1 Right Triangle Trigonometry
1.3
1.3 Applications and Solving Right Triangles
Throughout its early development, trigonometry was often used as a means of indirect measurement, e.g. determining large distances or lengths by using measurements o
114
Chapter 5 Graphing and Inverse Functions
5.2
Example 5.9
Find the amplitude and period of y = 3 sin x + 4 cos x.
Solution: This is sometimes called a combination sinusoidal curve, since it is the sum of two such
curves. The period is still simple to d
Trigonometric Functions of an Acute Angle Section 1.2
11. sin A =
3
4
12. cos A =
2
3
13. cos A =
15. tan A =
5
9
16. tan A = 3
17. sec A =
2
10
14. sin A =
7
3
13
2
4
18. csc A = 3
For Exercises 19-23, write the given number as a trigonometric function o
Properties of Graphs of Trigonometric Functions Section 5.2
113
Example 5.8
Find the amplitude and period of y = 2 sin ( x2 ).
Solution: This is not a periodic function, since the angle that we are taking the sine of, x2 , is not a
linear function of x, i
12
Chapter 1 Right Triangle Trigonometry
1.2
Example 1.10
Find the sine, cosine, and tangent of 75 .
D
Solution: Since 75 = 45 +30 , place a 306090 right triangle
ADB with legs of length 3 and 1 on top of the hypotenuse
of a 45 45 90 right triangle ABC w
112
Chapter 5 Graphing and Inverse Functions
5.2
Example 5.6
Find the amplitude and period of y = 3 cos 2 x.
Solution: The amplitude is | 3 | = 3 and the period is
2
2
= . The graph is shown in Figure 5.2.4:
y
3
2
3
1
6
0
6
1
3
2
2
3
5
6
x
7
6
4
3
3
2
5
3
Trigonometric Functions of an Acute Angle Section 1.2
11
We now know the lengths of all sides of the triangle ABC , so we have:
cos A =
csc A =
adjacent
5
=
hypotenuse
3
3
hypotenuse
=
opposite
2
sec A =
tan A =
opposite
2
=
adjacent
5
hypotenuse
3
=
adja
Properties of Graphs of Trigonometric Functions Section 5.2
111
Example 5.5
The period of y = cos 3 x is
shown in Figure 5.2.2:
2
3
and the period of y = cos
y = cos
y
1
2x
is 4. The graphs of both functions are
1
2x
y = cos 3 x
1
x
0
6
3
2
2
3
5
6
7
6
4