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 20
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π
2
−
=
1
sin
.
2
4
π
This function evaluation involved a familiar special angle,
, and so should be
4
performed by inspection ( as was done in the solution above). You can verify that the
end result is correct by calculating either
π
π
2
2
≅
≅
≅
≅
1 -
sin
. 0
7071067812
or
sin
. 0
7853981634
.
4
2
2
4
(
)
−
-1
Example: Find tan
.
3
(
)
−
-1
Solution: Set y = tan
3
. By the definition of the inverse tangent function,
π
π
=
−
−
<
<
tan
y
3
and
y
.
2
2
π
−
<
<
Since tan y is negative, y must lie in the interval
y
. 0
The angle
2
π
π
−
−
−
between
and
0
whose
tanget
is
3
is
, so
2
3
(
)
π
−
=
−
1 -
tan
3
.
3
Since this evaluation involved only a familiar special angle, it should be performed by
inspection. Of course, the result can be verified by calculation.
Problems:
In Problems 53-56, find the value of the inverse function without using a calculator.
1
3
1 -
54.
sin
1 -
55.
tan
2
3
-1
-1
56. cos
(-1)
57. sin
(2)
In Problems 57-59, find approximate values of the inverse trigonometric functions using a scientific
calculator. Round off the final answer to 4 (four) decimal places.
-1
58. cos
(-0.72)
-1
59. sin
(0.9901)
-1
60. tan
(-0.001)
Many applications involve compositions of trigonometric and inverse trigonometric functions such as
-1
-1
-1
sin(sin
x), sin
(sin x) and sin(cos
x). The simplest of these are of the form
-1
-1
-1
sin(sin
x), cos(cos
x) and tan(tan
x).
-1
-1
Consider the first of these. If y = sin
x, then x = sin y and hence, sin(sin
x) = sin y = x. Thus, as
long as x is in the domain of the inverse sine function,
- 20 -
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