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 22
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43π
π
3
5
1 -
1 -
73. Evaluate
74. Evaluate
tan
tan
.
sin
sin
.
4
6
π
π
3
25
1 -
1 -
75. Evaluate
sin
sin
.
76. Evaluate
cos
cos
.
5
13
-1
-1
Finally, we evaluate composition such as sin(cos
x) and tan(sin
x) of a trigonometric function with
some other inverse trigonometric function. These compositions cannot be evaluated directly from the
definitions of the inverse trigonometric functions.
4
1 -
Example: Evaluate
sin
cos
.
5
4
-1
Solution: Set y = cos
. We must find sin y. Since
5
4
cos y = 5
> 0, y is a quadrant I angle as shown
in Figure T40. By the definition of the cosine
4
a
a
=
=
=
function
cos
y
+
5
r
2
2
a
b
where (a , b) is any point on the terminal side
of angle y. To find sin y, we choose a point
(a , b) so that
a
4
=
.
+ b
5
2
2
Figure T40
a
The easiest choice is a = 4, so
+
=
+
=
2
2
2
a
b
16
b
5
+
=
2
16
b
25
=
±
b
. 3
since the terminal side of the angle is in quadrant I, b = +3. The point (a , b) = (4 , 3)
and, by the definition of the sine function,
4
3
=
=
1 -
sin
cos
sin
y
.
5
5
5
−
1 -
Example: Simplify
cos
tan
.
2
π
π
5
5
=
−
=
−
−
<
<
1 -
Solution: Set
y
tan
.
Hence,
tan
y
and
y
.
2
2
2
2
Since tan y is negative, y lies in the
π
−
interval
< y < 0 and y is a quadrant
2
IV angle.
Sketch angle y in standard position in
quadrant IV as shown in Figure T41.
Figure T41
- 22 -
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