Similar Triangles, Right Triangles, And The Definition Of The Sine, Cosine And Tangent Functions Of Angles Of A Right Triangle Worksheets With Answer Key Page 21
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43-1
sin(sin
x) = x
-1
-1
-1
-1
Similarly, if cos
x is defined, then cos(cos
x) = x. If tan
x is defined, then tan(tan
x) = x.
Problems:
-1
-1
61. Evaluate cos (cos
(-0.2119)).
62. Evaluate sin (sin
0).
-1
-1
63. Evaluate tan (tan
(0.9531)).
64. Evaluate sin (sin
(-3.95)).
-1
-1
-1
Simplifying compositions such as sin
(sin x), cos
(cos x) and tan
(tan x) requires more care.
-1
Example: Evaluate tan
(tan 1.07)).
π
π
−
<
<
=
1 -
Solution: Since 1.07 lies in the interval
x
,
tan
(tan
. 1
07
)
. 1
07
.
2
2
π
8
1 -
Example: Evaluate
cos
cos
.
7
Solution: The range of the inverse cosine function is the interval from 0 to π. Since
π
π
8
8
π
>
,
is not in the range of the inverse cosine function. Consequently,
7
7
π
π
8
8
≠
1 -
cos
cos
.
7
7
Our problem is to find an angle θ between 0 and π (in the range of cos
-1
) such that
π
8
θ
=
cos
cos
.
7
π
8
A quadrant II angle having the same reference angle as
will meet this requirement.
7
π
π
π
8
The reference angle for
is
. The quadrant II angle with reference angle
is
7
7
7
π
π
6
θ
π
=
−
=
.
7
7
π
π
π
8
6
6
-1
Since cos
= cos
and
is in the range of cos
.
7
7
7
π
π
π
8
6
6
=
=
1 -
1 -
cos
cos
cos
cos
.
7
7
7
Verify this value using a scientific calculator.
Problems:
-1
-1
65. Evaluate tan (tan
870).
66. Evaluate cos (cos
5.7).
-1
-1
67. Evaluate tan
(tan 1.4).
68. Evaluate tan
(tan(-0.6217)).
π
−
1 -
-1
70. Evaluate
cos
cos
.
69. Evaluate sin
(sin 0.0123).
4
(tan π).
-1
-1
71. Evaluate cos
(cos (-2π)).
72. Evaluate tan
- 21 -
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